Chapter 6: Series

Welcome to Last Minute Lecture.

This free chapter overview is designed to help students review and understand key concepts.

These summaries supplement not replaced the original textbook and may not be redistributed or resold.

For complete coverage, always consult the official text.

If you took a perfectly normal equilateral triangle and you just kept gluing smaller triangles to the middle of its edges and then you glued even smaller triangles to those new edges and like kept doing that forever, what would happen?

Oh, the cock snowflake.

Yeah, exactly.

Because logically, you're constantly adding length to the outside, so the perimeter must grow forever, right?

But mathematically, if you measure the area inside that spiky, infinitely complex shape,

well, it completely stops growing.

It holds a perfectly finite amount of space.

An infinite perimeter and closing a finite space.

I mean, it sounds impossible, like a glitch in reality or something.

It really does.

But it is entirely mathematically sound.

And honestly, it's just the beginning of what we are going to unpack today.

Welcome to the special Last Minute Lecture Deep Dive.

To you, our dedicated student listening right now, we are so glad you're here.

Today, we are basically hacking the source code of infinity.

We really are.

We're looking at the mathematics of series and sequences from chapter six of your Pure Math 1 course book.

And you know, our mission isn't just to hand you a list of formulas to memorize for an exam.

No, definitely not.

Well, we want to help you truly understand the mechanics behind them, right?

Exactly.

Because when you understand how sequences actually work, you aren't just doing algebra.

You're learning the language that dictates how bank loans compound or how search algorithms process data.

Or even how those beautiful, perfect spirals in nature actually grow.

Like, think of that image of the plant with the overlapping spiky green leaves that you might have seen at the start of the chapter.

Oh, yeah, the aloe plant looking thing.

Right.

It looks organic, but its growth is entirely governed by predictable mathematical patterns.

So today we're going to break down how to expand massively complex algebra, how to predict the exact future of a number sequence, and eventually how to add up an infinite number of things and somehow arrive at a completely finite answer.

Which is still wild to think about.

It is.

But to get to infinity, we need to start with the building blocks.

Before we even look at sequences of numbers, we have to look at sequences of algebraic terms.

Specifically, binomials.

Right.

By meaning two.

So expressions with two terms, like an A plus AB, all wrapped up in parentheses.

And the challenge begins when we raise those binomials to a positive integer power, which we'll call n.

So parentheses, A plus B parentheses, to the power of n.

Okay.

So if you expand A plus B squared, you just multiply A plus B by A plus B.

Exactly.

And you get a squared plus two AB plus B squared.

That's easy enough.

Yeah.

Standard foil method.

Right.

But as n gets larger, say A plus B to the fourth power or to the tenth, doing that multiplication manually becomes an absolute nightmare.

Just combining all those terms.

Hate.

I would definitely mess up a plus sign somewhere.

But there's a predictable pattern that saves us, right?

Thankfully, yes.

Let's look at a plus B to the fourth.

The very first term is just a plus A to the fourth.

And then in the next term, the power of A decreases by one to become A to the third, while the power of B steps into the spotlight, increasing from essentially B to the zero up to B to the one.

It's like a seesaw.

Yeah, exactly.

As the A power steps down, the B power steps up.

So you get A to the fourth, then a cubed B, then a squared B squared, then AB cubed.

All the way down until A is just gone and you only have B to the fourth.

And notice what happens to the total index in every single one of those terms.

If you have a cubed B to the first, three plus one is four.

And a squared B squared, two plus two is four.

The total power is always preserved.

It's so clean.

It really is.

That beautiful seesaw takes care of the variables, but we still have to figure out the numbers sitting in front of those variables, the coefficients.

And those coefficients follow a shape.

Pascal's triangle.

Yes.

Think of it built like a pyramid of bricks.

You start with a single brick at the very top with a one on it.

The next row down has two bricks, a one and a one.

And the third row has three bricks.

One, two, one.

Right.

Because every single brick in this pyramid rests on two bricks directly below it.

And the rule is so simple.

The number on any given brick is just the sum of the two bricks supporting it from underneath.

L 'Aise Pascal, the 17th century French mathematician,

he popularized the fact that this simple addition rule generates the exact coefficients needed for binomial expansions.

So if you want the coefficients for A plus B to the fourth, you just look at the fourth row of the triangle.

Exactly.

Which gives you one, four, six, four, one.

It's wild to think about how much math is just hiding in that simple pyramid too.

Like if you look down the diagonals, you find tetrahedral numbers.

The numbers that make 3D triangular pyramid.

Yeah.

And if you sum the shallow diagonals, you find the Fibonacci sequence, the one, one, two, three, five, eight pattern all just sitting there generated by one plus one.

It really highlights the interconnectedness of these mathematical structures.

It does.

But let me play devil's advocate here for a second.

Go for it.

Pascal's triangle is a brilliant tool if I need to expand a plus B to the fourth or maybe to the sixth.

I can draw that pyramid in the margins of my exam paper.

Sure.

But what if I'm designing a program and I need the expansion of A plus B to the 50th?

I cannot sit around drawing a 50 story pyramid of bricks just to find one coefficient.

We have to have a shortcut.

And mathematics provides one, thankfully.

We don't have to build the whole triangle to find a single brick.

We can use the binomial theorem.

Okay.

How does that work?

We use a specific tool denoted as N over R in tall parentheses, which is often read as N choose R.

You've probably seen it written as NCR on your scientific calculator.

Yes.

I've always wondered about that button.

So how does choosing relate to expanding algebra?

This is where we connect the what to the why.

When you were expanding a plus B cubed, you're really multiplying three brackets together, right?

A plus B times A plus B times A plus B.

Exactly.

To get a term like a squared B, you literally have to choose an A from two of the brackets and AB from the remaining one.

Oh, I see.

The mathematical question is simply,

how many different paths can you take to make that specific choice?

Oh, wow.

Okay.

So algebraic expansion is secretly just a problem of counting combinations.

We're just asking the math.

Out of N total brackets, how many ways can I choose exactly our number of B variables?

You've got it.

That is why the combinations formula, which is usually used in probability, perfectly predicts our algebraic coefficients.

That makes so much sense.

And to calculate N choose R manually,

we use factorial notation.

Right.

The exclamation point.

Exactly.

A factorial is a cascading multiplication.

So six factorial written as six exclamation point means you multiply six by every positive integer below it.

So six times five times four times three times two times one.

Which is 720.

So the formula for our specific brick in the pyramid, our coefficient N choose R, is N factorial divided by the product of R factorial and N minus R factorial.

Yes.

And this lets us launch a highly targeted strike to find one specific term without expanding the whole massive equation.

Okay.

Let's apply that targeted strike.

Imagine you're building an algorithm and you need to find the one constant number in a messy equation.

A term that won't change even if your variables do.

Ah, the term independent of X, like in the worked examples.

Exactly.

Suppose your expression is X minus two over X squared, all raised to the power of nine.

Okay.

So X minus a fraction.

Right.

Let me visualize this.

We have an X term and then a fraction with an X squared in the denominator.

To find a term completely independent of X, I need the X variables to completely cancel each other out.

Like a balancing scale.

Yeah, exactly.

On one side you have X's multiplying and on the other you have X's dividing.

And we have a total power of nine to distribute between them.

I need to tweak the powers until both sides have the exact same number of X's.

So how do you distribute that power of nine to balance the scale?

Let's see.

The second term has an X squared on the bottom.

It's heavier.

Right.

If I give the first term a power of six, that gives me X to the sixth multiplying on top.

That leaves the power of three for the second term, right?

Because six plus three is nine.

And if I raise that X squared in the denominator to the power of three, we multiply the powers.

Two times three is six.

That denominator becomes X to the sixth as well.

And the scale is perfectly balanced.

The X to the sixth multiplying on top and the X to the sixth dividing on the bottom annihilate each other.

Leaving only a constant number.

Exactly.

And to find out exactly what that number is, you just calculate the coefficient using nine choose three and multiply it by whatever constants were attached to your X's in this case, the negative two that was being cubed.

It's a beautifully efficient process.

I love when the mechanics just click into place like that.

Me too.

Okay.

So we've mastered how terms progress in complex algebraic expansions.

Let's shift gears entirely.

Let's do it.

Let's look at numbers that progress in a straight line.

We're moving to linear number sequences known as arithmetic progressions or APs.

An arithmetic progression is defined by a very specific rhythm.

Each term differs from the one before it by a constant fixed amount.

We call this the common difference, right?

Denoted with the letter D.

Right.

And the starting point, the very first term of the sequence is always called A.

Okay.

So if I have a sequence of five, eight, 11, 14,

my first term A is five.

I'm taking a step of three every time.

So my common difference D is three.

Which means we can predict any future number in that sequence.

To find the Nth term, you start at your foundation A and you add that difference D.

But how many times do you add it?

Well, if you want the 10th term, you don't take 10 steps.

You are already standing on the first term, so you only need to take nine steps.

Oh, so it's always one less.

Exactly.

Therefore, the universal formula for the Nth term is A plus parenthesis N minus one times D.

That's the power of predicting a sequence.

But things get tricky when we want to add all those terms together, right?

Finding the sum of an AP.

It does get more complicated.

There's a legendary historical story from the textbook about this involving a seven -year -old Carl Gauss.

Oh, I love this story.

His teacher wanted a break, so he told the class of young children to add up all the whole numbers from 1 to 100.

The teacher figured this would keep them quiet for an hour.

But Gauss walked up almost immediately with the correct answer.

50 -50.

Yeah.

He realized something fundamental about the symmetry of linear sequences.

He didn't just add 1 plus 2 plus 3, grinding through the arithmetic.

No, he essentially folded the sequence in half like a sandwich.

Right.

He paired the very first number, 1, with the very last number, 100.

Together, they make 101.

And then he paired the second number, 2, with the second to last number, 99.

They also make 101.

3 and 98 make 101.

And because he is dealing with 100 numbers, folding them to pairs means he has exactly 50 pairs.

50 pairs, each summing to 101.

Multiply them together and you get 50 -50.

It's such an elegant conceptual leap for a kid.

It really is.

Now let's see if we can map Gauss's mental trick onto our algebra to build a universal formula for the sum of any arithmetic sequence which we call s sub n.

Okay.

Pairing the first and last numbers.

Algebraically, our starting term is a, and let's call our very last term l.

So Gauss's pair of 101 is represented by a plus l.

That's the value of every single pair.

But how do we prove algebraically how many pairs we have without writing all the numbers out?

Let's imagine writing the entire sum out as a long equation.

S sub n equals the first term a plus the next term plus the next all the way to l.

Okay, I'm picturing it.

Now underneath it, let's write the exact same equation but entirely backwards.

Start with l then work all the way down to a.

So I have two equations stacked vertically.

Right.

Now add them together straight down.

S sub n plus s sub n gives me 2s sub n on the left side.

And on the right.

The first vertical pair is a plus l.

The next vertical pair also simplifies to a plus l.

Every single column adds up to a plus l.

And since there are n terms in my sequence, I have exactly n of these a plus l pairs.

Now your equation is 2s sub n equals n times a plus l.

I divide both sides by 2 and there it is.

S sub n equals n over 2 times a plus l.

The number of terms divided by 2 multiplied by the sum of the first and last term.

Gauss's sandwich trick?

Proven with variables.

And if you don't happen to know the last term l, you just substitute in our nth term formula a plus n minus 1d.

Exactly.

That gives you the alternative version.

S sub n equals n over 2 times bracket 2a plus n minus 1d bracket.

It's the exact same logic, just packaged differently depending on what information a problem gives you.

Precisely.

Okay.

So adding a constant d gets us that steady linear climb.

But what if we change the rules?

How so?

What if instead of adding a constant, we multiply by a constant?

We are leaving arithmetic progressions behind and entering the world of geometric progressions, or GPs.

This fundamentally changes the growth rate.

A geometric progression is exponential.

Instead of a common difference, we have a common ratio denoted by r.

So the sequence starts with a, then you multiply by the ratio to get a r, multiply again for a r squared, then it r cubed, and so on.

Think of a viral video where one person shares it with three friends and each of them shares it with three friends.

It explodes outwards.

The nth term formula makes sense here.

It's just the first term a multiplied by the ratio r raised to the power of n minus 1.

Again, we multiply by the ratio one less time than the position of the term.

But finding the sum of this explosion, that seems way harder.

Gauss's reversing and adding trick worked beautifully for arithmetic because the steps were even.

If I try to stack and add a geometric series, nothing lines up.

It's a mess of different powers.

Right.

Reversing won't work here.

We need a new algebraic derivation.

We use the shift and subtract trick.

Shift and subtract.

Okay.

Let's ground this in concrete numbers first.

Imagine your sequence is 1 plus 2 plus 4 plus 8.

The ratio is 2.

What happens if you multiply that entire sum by the ratio?

Every number doubles.

So my new sequence is 2 plus 4 plus 8 plus 16.

Look at the two sequences.

The first is 1, 2, 4, 8.

The second is 2, 4, 8, 16.

Notice how the middle numbers 2, 4, and 8 are completely identical in both.

Oh, yeah.

If you subtract the first sequence from the second sequence, all those matching middle numbers completely vanish.

The 2 cancels the 2.

The 4 cancels the 4.

The 8 cancels the 8.

I'm left with just the 16 from the new sequence minus the 1 from the original sequence.

Exactly.

Now apply that concrete logic to the abstract algebra.

Yes, let's try.

Let your original sum be equation 1.

S sub n equals a plus ar plus ar squared up to ar to the n minus 1.

Got it.

Now multiply the whole equation by the ratio r.

Equation 2 becomes r times s sub n equals ar plus ar squared plus ar cubed all the way up to ar to the n.

So equation 2 is basically equation 1, but every single term has been shifted up by 1 power of r.

So we subtract equation 1 from equation 2.

On the left side of the equal sign, we have r times s sub n minus s sub n.

And on the right side, the dust settles.

Just like in our number example, all of the messy middle terms cancel out.

Only two terms survive the subtraction.

The mass of ar to the n from the second equation minus the starting a from the first equation.

OK, so r times s sub n minus s sub n equals ar to the n minus a.

If I factor out the s sub n on the left, I get s sub n times r minus 1.

And if you factor out the a on the right.

I get a times r to the n minus 1.

I just divide by r minus 1 to isolate the sum.

And boom, the final formula, s sub n equals a times r to the n minus 1 all divided by r minus 1.

That is deeply satisfying.

It's a really powerful tool for calculating compound interest, population growth, or any exponential system.

Wait, I do have a practical question though about this.

Sure.

I've noticed in the text, there are two versions of this sum formula.

There's the one we just arrived with r to the n minus 1 on top.

But there's also s sub n equals a times 1 minus r to the n divided by 1 minus r.

The terms in the parentheses are just flipped.

Do they do different things?

They do the exact same thing.

Really?

Why have two then?

It is purely a matter of mathematical hygiene, honestly.

It's to avoid dealing with negative denominators.

Oh, I see.

Think about it.

If your ratio r is a large number, say 5, you want to use the first formula r minus 1 because 5 minus 1 gives you a positive 4 in the denominator.

Right.

But if my sequence is shrinking and my ratio r is a fraction, like 0 .5, doing 0 .5 minus 1 gives me a negative 0 .5.

It just invites arithmetic mistakes on a test.

So instead, I use the flipped version.

1 minus r, 1 minus 0 .5 keeps the denominator positive.

Work smarter, not harder.

Precisely.

And that idea of a shrinking sequence where r is a fraction brings us to the final and perhaps most mind -bending concept.

Infinity.

Infinity.

We found the sum of a specific number of terms, n.

But what if n never stops?

What if we want to add up an infinite number of terms?

OK, let me be the skeptic here.

If I keep adding numbers forever and ever, shouldn't the answer always just explode into infinity?

Think of the old wall paradox.

If I stand in a room and step halfway to the wall, then take another step that covers half of the remaining distance and half again, and I keep doing that forever.

I am taking an infinite number of steps.

So shouldn't the total distance I travel be infinite?

Your intuition is screaming that infinity equals infinity, but it misses the mechanical reality of fractions.

Look closely at the math of your wall example.

Your starting distance is a half.

Your ratio is a half.

Every step you take is smaller than the last.

You add a half, then a quarter, then an eighth, a sixteenth.

The terms you are adding shrink incredibly fast.

Right, because a half times a half is a quarter times a half is an eighth.

The denominator just gets massive, which means the fraction itself gets microscopic.

So what happens to our geometric sum formula as the number of terms, n, approaches infinity?

The crucial part of the formula is that r to the n term.

Yeah.

If r is a fraction strictly between negative one and one, taking it to an infinitely large power makes it infinitely small.

It gets closer and closer to exactly zero.

It effectively vanishes.

It vanishes completely.

We say that this n tends to infinity, r to the n tends to zero.

And when that happens, that clunky sum formula simplifies down to something incredibly elegant.

Because the one minus r to the n part just becomes one minus zero, which is just one.

Exactly.

So the formula for the sum to infinity, s sub infinity, distills down to simply the first term, a, divided by one minus r.

That is it, a perfectly finite limit.

So you are taking an infinite number of steps, but you will never walk further than the wall.

Right.

Let's visualize this another way from the book.

Imagine a square that is two meters by two meters.

Its total area is four.

Now, I put a rectangle inside it with an area of two.

Then I add a square with an area of one, then a rectangle of a half, a quarter, an eighth.

I can keep drawing smaller and smaller shapes inside that boundary forever.

An infinite number of shapes, but they will perfectly fill and never exceed that finite area of four.

It provides a concrete anchor for an abstract concept.

We call this a convergent series.

And we actually use convergent infinite series every single day.

We do.

Yeah.

Recurring decimals.

Take the fraction one -third as a decimal.

It's 0 .3333 going on forever.

It looks like a single number, but it's actually an infinite geometric series.

Of course.

Three -tenths plus three -hundredths plus three -thousandths.

The first term, a, is 0 .3.

The common ratio, r, is 0 .1.

If you plug those into our sum -to -infinity formula, 0 .3 divided by 1 minus 0 .1, you get 0 .3 divided by 0 .9, which simplifies to exactly one -third.

It justifies the decimal conversions we've been taking for granted since primary school.

And I love the history here, too.

Simon Stephen, way back in 1585,

was the mathematician who popularized the use of infinite decimals.

Imagine grappling with the concept of infinity without modern calculus to back you up.

Speaking of grappling with infinity, let's return to the provocation you started this deep dive with.

The cock snowflake.

Right, the fractal.

Starting with a triangle and constantly adding smaller triangles to the edges, infinitely.

If you apply the series formulas we've discussed today to the geometry of that snowflake, you discover the mathematical proof behind the paradox.

Because you're constantly adding length to the edges, the geometric series that represents the perimeter has a ratio greater than one.

So therefore, it diverges.

The perimeter truly is incident.

The series that represents the area those new triangles are adding, that series has a fractional ratio.

It converges.

The math definitively proves it encloses a totally finite area, exactly eight -fifths of the original triangle's area.

An infinite, never -ending perimeter holding a finite, contained space.

It forces us to redefine our intuitive understanding of space and dimension.

And it is a perfect example of why the rigorous mathematics of series, moving past what feels right and trusting the algebra is so deeply necessary.

We've covered a massive amount of conceptual ground today.

We started by expanding complex algebraic binomials, realizing that combining variables is really just a problem of counting paths, guided by Pascal's pyramid and factorial notation.

Then we moved to linear arithmetic progressions, summing them up using Gauss's brilliant symmetry.

From there, we explored the exponential growth of geometric progressions, mastering the shift and subtract derivation.

And finally, we stared into the face of infinity, proving that an infinite number of shrinking fractions can indeed add up to a perfectly finite answer.

If you can grasp these mechanics,

you're doing so much more than just prepping for your pure mathematics one exam.

You're learning the structural language of the world around you.

On behalf of the entire last -minute lecture team, we want to say thank you for trusting us with your study prep.

And hey,

keep practicing those derivations, understanding the why behind Gauss's trick or the shift and subtract method makes the how so much easier when you are staring at a blank exam paper.

Before we go, I want to leave you with one final thought to mull over.

It's about a mathematician named George Cantor.

Cantor formalized many of the ideas we have about infinity today, but he proved something deeply unsettling.

Not all infinities are equal.

Wait, really?

Yeah.

He proved mathematically that the infinite set of all real numbers, every decimal and fraction, is actually larger than the infinite set of just whole numbers.

An infinity that is quantifiably bigger than another infinity.

So as you close your books today and look back at that plant with its perfect mathematical spirals, ask yourself this.

If there are different sizes of infinity in pure mathematics, what does that imply about the infinite bounds of the actual physical universe you live in?

Good luck with your studies.

We'll see you on the next Deep Dive.