Sequences and series review
Sequences and series (part 2) Finding the sum of an infinite geometric series.
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- Welcome back.
- So where we left off in the last video, I'd shown you
- this thing called the geometric series.
- And, you know, we could have some base a.
- It could be any number.
- It could be 1/2, it could be 10.
- But that's just-- but some number.
- And we keep taking it to increasing exponents, and we
- sum them up, and this is called a geometric series.
- And so I want to figure out the sum of a geometric series of,
- you know, when I have some base a, and I go up to some
- number a to the n.
- What-- is this a to the-- why did I write
- a to n minus 2 there?
- That should be a to the big N.
- My brain must have been malfunctioning in
- the previous video.
- That always happens when I start running out of time.
- But anyway.
- Let's go back to this.
- So I defined s as this geometric sum.
- Now I'm going to define another sum.
- And that sum I'm going to define as a times s.
- And that equals-- well, that's just going to be a times
- this exact sum, right?
- And that's the same a as this a, right?
- That a is the same as this a.
- So what's a times this whole thing?
- Well, it's the a times a to the zero is-- let me
- write it down for you.
- So this'll be a because I just distribute the a, right? a
- times a to the zero, plus a times a to the 1, plus a times
- a squared, plus all the way a times a to the n minus one,
- plus a times a to the n.
- I just took an a and I distributed it along
- this whole sum.
- But what is this equal to?
- Well, this is equal to a times a to the zero.
- That's a one-- a to the first power-- plus a squared, plus a
- cubed, plus a to the n, right?
- Because you just add the exponents, a to the n.
- Plus a to the n plus 1.
- So this is as.
- And we saw before that s is just our original sum.
- That is just a to the zero, plus a to the 1, plus a
- squared, plus up, up, up, up.
- All the way to plus a to the n, right?
- So let me ask you a question.
- What happens if I subtract this from that?
- What happens?
- If I say, as minus s.
- Well, I subtracted this from here, on the left hand side.
- What happens on the right hand side?
- Well, all of these become negative, right?
- Let me do it in a bold color.
- This becomes-- because I'm subtracting-- negative,
- negative, these are all negatives.
- Negative.
- Negative.
- Well, a to the first, minus a to the first.
- That crosses out. a squared minus a squared crosses
- out. a to the third, it'll all cross out.
- All the way up to a to the n, right?
- So what are we left with?
- We're just left with minus a to the zero, right?
- We're just left with that term.
- And we're just left with that term.
- Plus a to the n plus 1.
- And of course, what's a to the zero?
- That's just 1.
- So we have a times s minus s is equal to a to
- the n plus 1 minus 1.
- And now let's distribute the s out.
- So we get s times a minus 1 is equal to a to the n
- plus 1 minus 1, right?
- And then what do we get?
- Well, we can just divide both sides by a minus 1.
- Let me erase some of this stuff on top.
- I think I can safely erase all of this, really.
- Well, I don't want to erase that much.
- I want to erase this stuff.
- That's good enough.
- OK.
- So I have just-- dividing both sides of this equation by a
- minus 1, I get s is equal to a to the n plus 1 minus
- 1 over a minus 1.
- So where did that get us?
- We defined the geometric series as equal to the sum.
- From k is equal to 0, to n of a to the k.
- And now we've just derived a formula for what that
- sum ends up being.
- Equals a to the n plus 1 minus 1 over a minus 1.
- And why is this useful?
- We now know, if I were to say, well, what is-- let me clean
- up all of this, as well.
- Let me clean up all of this and we can-- OK.
- So if I said, you figure out the sum of, I don't know, the
- powers of 3 up to 3 to the, I don't know, 3 to
- the tenth power.
- So, you know, 3.
- So 3 to the zero, plus 3 to the one, plus 3 squared, plus all
- the way to 3 to the tenth.
- So this is the same thing as the sum of k equals zero
- to 10, of 3 to the k.
- Right?
- So this formula we just figured out, a is 3 and n is 10.
- So this sum is just going to be equal to 3 to the eleventh
- power minus 1 over 3 minus 1.
- Which equals-- well, I don't know what 3 to
- the eleventh power is.
- Minus 1 over 2.
- So that's kind of useful.
- That is a number.
- Although you'd have to memorize your exponent tables to the
- eleventh power to do that.
- But I think you get the idea.
- This is especially useful if we were dealing with-- well, if
- the base was a power of ten, it would be very, very easy.
- But what I actually want to do now is I want to take this and
- say, well, what happens if n goes to infinity?
- Let me show you.
- So what happens?
- So there's two types of series that we can take-- that's
- not what I wanted to do.
- There are two types of series that we can take that we
- can find the sums of.
- There's finite series, and infinite series.
- And in order for an infinite series to come up to a sum
- that's not infinity, they need to-- what we say--
- they need to converge.
- And if you think about what has to happen for them to converge,
- every next digit has to essentially get smaller and
- smaller and smaller, as we go towards infinity.
- So let's say that a is a fraction.
- a is 1/2.
- So how does a geometric series look like if we have 1/2 there?
- So let's say that we're taking the geometric series from k
- is equal to 0 to infinity.
- So this is neat.
- We're going to take an infinite sum, an infinite number of
- terms, and let's see if we can actually get an actual number.
- You know, we take an infinite thing, add it up, and it
- actually adds up to a finite thing.
- This has always amazed me.
- And the base now is going to be 1/2.
- It's 1/2 and it's going to be 1/2 to the k power.
- So this is going to be what?
- 1/2 to the zero, plus 1/2, plus-- what's 1/2 squared?
- Plus 1/4, plus 1/8, plus 1/16.
- So as you see, each term is getting a lot, lot smaller.
- It's getting half of the previous term.
- Well, let's say, what happens if this wasn't infinity?
- What happens if this was n?
- Well, then we'd get plus 1 over 2 to the n, right?
- 1/2 to the n is the same thing as 1 over 2 to the n.
- And if we look at the formula we figured out, we would say,
- well, that is just equal to 1/2 to the n plus 1, minus
- 1, over 1/2 minus one.
- And that would be our answer.
- We'd have to know what n is.
- But now we want to know what happens if we go to infinity.
- So this is essentially a limit problem.
- What happens-- what's the limit, as n goes to infinity,
- of 1/2 to the n plus one minus 1 over 1/2 minus 1?
- Well, all of these are constant terms, so nothing happens.
- So what happens as this term, right here, goes to infinity?
- What's 1/2 to the infinity power?
- Well, that's zero.
- That's an unbelievably small number.
- Take 1/2 to arbitrarily large exponents, this just goes to 0.
- And so what are we left with?
- We're just left with this equals minus 1 over 1/2 minus
- 1, or we could multiply the top and the bottom by negative 1.
- And we get 1 over 1 minus 1/2.
- Which equals 1 over 1/2, which is equal to 2.
- I find that amazing.
- If I add 0 plus 1/2 plus 1/4 plus 1/8 plus 1/16 and I never
- stop-- I go to infinity-- and not infinity, but I go to 1
- over essentially 2 to the infinity-- I end up with
- this neat and clean number.
- 2.
- And this might be a little project for you, to actually
- draw it out into like maybe a pie and see what happens as
- you keep adding smaller and smaller pieces to the pie.
- But it never ceases to amaze me, that I added an infinite
- number of terms, right?
- This was infinity.
- And I got a finite number.
- I got a finite number.
- Anyway, we ran out of time.
- See you soon.
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