0:00:00.508,0:00:04.975
One type of series where we can [br]actually come up with a formula

0:00:04.975,0:00:07.991
for the sequence of partial sums

0:00:07.991,0:00:10.024
is called a geometric series.

0:00:10.024,0:00:12.974
You might remember learning [br]about geometric sequences

0:00:12.974,0:00:14.640
in the previous section.

0:00:14.640,0:00:18.774
Basically, it's the same idea except [br]the terms are being added together.

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Every term is being multiplied [br]by a common ratio, which we call r,

0:00:24.774,0:00:31.474
and that means the formula [br]for our generic nth term would be ‘a’,

0:00:31.474,0:00:33.324
which was the first term,

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and then times r to the [br]n minus 1st power,

0:00:36.790,0:00:41.724
so that would be the formula for [br]our series for each individual term.

0:00:41.724,0:00:44.907
But the idea here is, in some cases,

0:00:44.907,0:00:49.073
we can find a formula [br]for the nth partial sum,

0:00:49.073,0:00:52.040
for the sum of the first n terms, [br]and based on that,

0:00:52.040,0:00:56.207
we can determine whether [br]this series will converge or diverge.

0:00:56.207,0:00:59.657
A couple of cases to get [br]out of the way first,

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if r is equal to 1, [br]what would this thing look like?

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Well, what it would look like is [br]just ‘a’ and then plus another ‘a’

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and then plus another ‘a’ and [br]continuing in that fashion forever.

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Now, if I just added up [br]the first n terms here,

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I would have a total of n a’s [br]all being added together.

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That would be n times ‘a’ [na].

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But if we take a limit as n [br]approaches infinity for n times ‘a’,

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that limit is going to be infinite.

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So if r is equal to 1, [br]this series is going to diverge.

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Hopefully, that makes sense;

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if you add up an infinite number of [br]terms and they're all staying the same,

0:01:52.809,0:01:56.022
your sum is not going [br]to approach a value.

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Basically, the only way [br]this wouldn't diverge

0:01:58.589,0:02:00.139
is if ‘a’ is equal to 0,

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and most of the time, [br]we're going to ignore that case

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because it's really not a very interesting [br]version of a geometric series.

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Next up, we're going to look [br]at the case: if r is not equal to 1.

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In this case, the sum of the first n terms

0:02:16.416,0:02:19.215
would just be all of these [br]terms added together.

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So this is our nth term.

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But actually, we can do something [br]interesting algebraically here,

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if directly below this, [br]I take r times all of the terms,

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because if I do that, [br]I take r times this term,

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I'm going to get ‘a’ times r.

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If I take r times this term, [br]I'll get ‘a’ times r squared.

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I'm going to continue in this fashion,

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and this is actually going to [br]be the second-to-last term

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because that's what I would get [br]if I took the previous term times r.

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That means, if I take this times r, [br]I'm going to get ‘a’ times r to the n.

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And it turns out, since both [br]of these sums are finite,

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(there's a set number of terms),[br]I can combine them together.

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But surprisingly, [br]I'm actually going to subtract them.

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I'm going to get Sn minus r times Sn.

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My goal here is to come up [br]with a formula for just Sn.

0:03:18.949,0:03:21.732
But the interesting thing that happens is,

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if we were to subtract all of [br]these terms from these terms,

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almost all of the terms [br]are going to cancel out.

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The only ones that are going [br]to be left are the very first term

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and the very last term of this one.

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Since we're solving for Sn,

0:03:38.786,0:03:46.285
it hopefully makes sense to factor [br]that out from this side of the equation.

0:03:46.285,0:03:48.116
There's also a common factor of ‘a’

0:03:48.116,0:03:53.749
that we could factor out [br]from this side of the equation.

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Solving for Sn, we can get this [br]by itself by dividing by 1 minus r.

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The reason we don't have [br]to worry about dividing by 0

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is because we're working [br]under the assumption

0:04:04.849,0:04:17.316
that r is not equal to 1 in this case, [br]which gives us this formula.

0:04:21.530,0:04:23.582
What we're looking at here

0:04:23.582,0:04:28.842
is a formula for the sum of the first [br]n terms of a geometric series

0:04:28.842,0:04:32.224
as long as our common ratio is not 1.

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What I'm interested in, though,

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is if the infinite series is [br]going to converge or not.

0:04:37.141,0:04:39.109
Since we have a formula for Sn,

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now we're going to take the limit [br]of this as n approaches infinity

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to see what's going to happen to the sum.

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One thing you'll maybe notice here is,

0:04:56.937,0:05:01.786
there's only one part of this[br]entire formula that has an n in it,

0:05:01.786,0:05:04.154
and it's this part right here.

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This is actually something [br]we talked about in Section 11.1.

0:05:08.587,0:05:16.954
This right here is only going to converge [br]if the absolute value of r is less than 1.

0:05:16.954,0:05:19.769
Think about it: If this is greater than 1,

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then multiplying it [br]by itself multiple times

0:05:22.554,0:05:26.037
will cause it to basically [br]blow up to infinity.

0:05:26.037,0:05:29.637
And if it's less than negative 1, [br]the same problem is going to happen,

0:05:29.637,0:05:32.970
except the sign will be alternating [br]from positive to negative.

0:05:32.970,0:05:38.303
So this is only going to converge [br]if the absolute value of r is less than 1.

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Any other value for r, [br]and this thing is going to diverge,

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except of course, 1, [br]which we've already ruled out.

0:05:45.783,0:05:51.613
Basically, this sequence, which stands[br]for the sequence of partial sums,

0:05:51.613,0:06:00.797
converges if the absolute [br]value of r is less than 1;

0:06:00.797,0:06:09.930
and it's going to diverge if the absolute [br]value of r is greater than or equal to 1.

0:06:10.912,0:06:13.358
We've already excluded this case,

0:06:13.358,0:06:17.174
but if r was negative 1, [br]we would run into the same problem.

0:06:17.174,0:06:19.074
It would still diverge.

0:06:19.074,0:06:21.107
What does it converge to?

0:06:21.107,0:06:24.141
Well, if the absolute [br]value of r is less than 1,

0:06:24.141,0:06:29.561
then this term right here is [br]going to approach 0 as n grows.

0:06:29.561,0:06:31.890
And that means, [br]what we're going to be left with

0:06:31.890,0:06:36.373
is just ‘a’ in the numerator [br]divided by 1 minus r,

0:06:36.373,0:06:40.190
and that winds up being [br]the sum of a geometric series.

0:06:40.190,0:06:46.257
If the absolute value of r is less than 1,

0:06:46.257,0:06:54.924
then this series (the geometric series) [br]is going to have a sum,

0:06:54.924,0:07:02.957
and that sum is going to equal [br]‘a’ divided by 1 minus r.

0:07:02.957,0:07:04.974
If this condition is not met,

0:07:04.974,0:07:07.691
then the geometric series [br]is going to diverge.[br]