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Find the nth term of the given infinite series for which $n=1,2,3, \dots$$$\frac{1}{\sqrt{2}}-\frac{1}{2}+\frac{\sqrt{2}}{4}-\frac{1}{4}+\cdots$$

Calculus 2 / BC

Chapter 30

Expansion of Functions in Series

Section 1

Infinite Series

Series

Missouri State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Lectures

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In mathematics, integratio…

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01:28

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01:03

Find the sum of each geome…

00:35

Find the first four terms …

01:58

03:51

00:44

Find the first five terms …

Write the first four terms…

01:27

Find a formula for the $n$…

03:23

nth Term of a Sequence Fin…

00:56

or we want to identify the term of the infinite series for which we have N equals 123 And the series is given as one over root two minus one, half its route to over four minus 1/4 and so on. This question is challenging understanding of sequences and series in particular challenges to identify the pattern that's present in this particular series. To identify A. M. So we have that are individual terms are a one equals one over root two or route to the negative first, A two is negative 1/2 or negative to the negative second. A three is negative route to over four, which is another way of saying negative route to the 3rd and 1/4 to find similarly. Thus we see that each of our A. M. Has a route to raise some negative power. And so now that we have identified the pattern, we see that cluster minus or a sign of alternating and the absolute value of A. M. Is always route to the negative. And so we must have A. M. Equals negative one to the endless one times one over root two to the end.

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