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# Calculate the sum of the series $\sum_{n = 1}^{\infty} a_n$ whose partial sums are given.$s_n = 2 - 3(0.8)^n$

## $$2$$

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##### Kristen K.

University of Michigan - Ann Arbor

##### Michael J.

Idaho State University

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### Video Transcript

Let's go ahead and calculate the sum of this Siri's who's partial sums air given by the following formula over here. So let's distinguish between the N and the Essen. So recall by definition, this which is the and partial some that's our notation. That ass notation is further the partial. Some and the end is telling you how many terms or in the sun and this is the fines and be a one. Let's write it this way. The sun from Let's go k equals one to end of a K. So this is just the sum of the A's. But on Liam to some a m. So now and our problem there telling us what s it is they're saying if you add the first and numbers, you get to minus three, zero point eight to the end and now we want the infinite something so here and equals one to infinity of a M. That's equals the limit. Let's write it this way, since we're there using end here, let's write it as Argos to infinity and equals one. So are of I am Caesar that this right here. This is just as our so lim as our goes to infinity of s are so that's two minus three point A to the O. R. And his fueling our goto infinity and take the limit. This term right here goes to zero because point eight. So the R goes to zero as our goes to infinity and the way to see that his point is between zero and one. So each time you multiply by point, you're getting smaller and smaller. And so here this will go to zero and we're just left with two, and that's our final answer.

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##### Kristen K.

University of Michigan - Ann Arbor

##### Michael J.

Idaho State University

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Join Bootcamp