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JH
Numerade Educator

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Problem 1 Easy Difficulty

(a) What is an alternating series?
(b) Under what conditions does an alternating series converge?
(c) If these conditions are satisfied, what can you say about the remainder after $ n $ terms?

Answer

a) An alternating series is a series whose terms alternate between the positive sign and the negative sign.
b) An alternating series converges $\sum a_{n}$ under the following two conditions:
(1) $\left|a_{n+1}\right| \leq\left|a_{n}\right|,$ and $(2) \lim _{n \rightarrow \infty}\left|a_{n}\right|=0$
c) The remainder after $n$ terms should be no more than absolute value of the very first term right after the $n$ th term. That is, $\left|R_{n}\right| \leq\left|a_{n+1}\right|$

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Video Transcript

from party, eh? And all. Turning Siri's is the theories you write this out is a song whose terms alternate between positive and negative. So that's what all straining Siri's is for Part B. First, we should note that on alternating Siri's could always be written in the form this form right here or usually. Sometimes people like to write in this form, depending on the Siri's. So here, what we need is for being to be digging his ear off. We need that the limit of being a zero, and we need that The pianist are decreasing for all in. So here are conditions. So let's call these A, B and C are those already being used like he's a different amuse numbers here. Here's one two in three. So this is the answer for party under what conditions will on alternating Siri's converge well and also ending Syria's will converge when one through three are satisfied. So that's our answer for part two, and after part three, let's go ahead and look at the remainder. So if these conditions are satisfied, so that's assuming that the alternating Siri's converges. So here we have our ulcer, any Siri's, but since it doesn't matter what the starting point is, and this entire sum is equal to some number US. Bye, Assumption. We're saying that it converges from party now the remainder, after adding in terms, is given by s minus the impartial some. And this is by a theorem in your textbook alternating Siri's estimation there, Um, this says that the remainder satisfies this, but we know that in the limit that limit beyond equal zero. So this implies that the limit of our end equal zero. So this tells us that in the limit that the remainder, after adding in terms, goes to zero, and that's our final answer.