Original subtitles

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Today we're going to be talking about how to find the sum of the series of partial sums.

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And in this particular problem we've been given a series of partial sums as Subhan is equal to two minus

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three times 0.8 raised to the power.

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Now before we talk about how to find the sum of the series I want to talk briefly about the difference

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between a series and a series of partial sums just to illustrate this concept quickly.

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Let's say that we have a regular series we'll call it a Subhan and we're going to go ahead and list

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the terms in this series and then I want to show you the difference between a regular series and the

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partial sums of the series partial sums are usually denoted by Subhan to indicate.

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As for some.

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So if the terms in our series for example are just one two three four five six right like this those

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are the terms in our regular series the series of partial sums will be the sum of the associated term

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and every value in the series before it.

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So for the first term it's just exactly the same right.

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We have one here we have one here.

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But for our second term instead of just two we take two and we added to the value of the series before

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that which is once a two plus one is three for the third term.

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We take the third term and a second term.

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And the first term and add those together three plus two plus one is six.

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We get six there.

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When we add 4 to that we get 10.

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We had five to 10 we get 15 and we had six to 15 and we hit 21.

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This is the series of partial sums of the series a sub n so we would denote this series 1 2 3 4 5 6

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as the series A in the series of partial sums of this series would be denoted as Subban as 1 3 6 10

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15 21.

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So that's the difference and in this particular case obviously we have this series of partial sums.

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Esteban in this equation here.

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Now how do we find the sum of the series of partial sums.

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Well what we need to know is that what we're really talking about here is the infinite sum of the series

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a sub Subban right from end equals one to infinity.

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That's the sum of the series.

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But we're dealing here with partial sums.

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Well it turns out that the sum of the series as we take an out to infinity is just equal to the limit

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as and goes to infinity of as sub n the series of partial sums.

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So this is a convenient tool for us as a convenient equation for us to you often because if we have

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the series of partial sums here then we can just take the limit as that series goes to infinity and

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it gives us the sum of our original series.

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So what we're going to be evaluating here is the limit as and goes to infinity of s n.

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Now to do that in place of S N we can just plug in our equation here.

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So we'll get the limit as and goes to infinity of our equation to minus three times 0.8 raised to the

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end power.

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Now if we essentially plug in an infinite value here for n if we plug in an extremely large value for

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and what we find is that this quantity here 0.8 raised to the power continues to get smaller and smaller

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and smaller and smaller because if you take point eight and you multiply it by itself what you get of

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course is point 6 4 which is a smaller value than point eight.

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If you multiply it again by point eight point six four by point eight or in other words point eight

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cubed you get Point 5 1 to the value continues to decrease and get less and less and less and less.

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So what we can say is that goes to infinity this point eight to the n becomes zero and it's just going

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to go away it's going to become 0.

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So what we're left with here is two minus three times zero.

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Of course that takes away the three as well we get three times zero here that goes away.

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And we're just left with a value of two.

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What this tells us is that the sum of the series the sum of the series a Sabahan whatever it is and

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we didn't have the series a seven to model we only had seven.

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But the sum of the series is two.

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And we know that using the equation for the series of partial sums which we denoted as 7.

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So that's how we use the series of partial sums to find the sum of the series a 7.

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In this case two.