How to find the sum of an alternating series? Ask Question Asked 12 years, 10 months ago Modified 1 year, 4 months ago

Jun 18, 2017 · 4 I want to find the sum of $$\sum_ {n=1}^ {\infty}\frac { (-1)^ {n+1}} {n^2}$$ I know that this is equal to $\frac {\pi^2} {12}$ thus I was thinking this must just be a taylor series of …

Proving that this series converges can be done using the alternating series test: any series that alternates forever between positive and negative terms, where each term is smaller than the …

Can you use the formula $ (x+1)^2=x^2+2x+1$ to expand $ (x^2+1)^2$? If so, a similar operation on your first sum will give the second.

Mar 25, 2022 · The series of odd terms is the Leibniz Formula for $\pi$, whose sum is $\frac\pi4$, and the series of even terms is $-\frac12$ times the Alternating Harmonic Series, whose sum …

Nov 8, 2018 · This same equation doesn't work for an alternating geometric series such as $ (-2^n)$, where the series is $1,-2,4,-8,16$. I'm looking to find the summation of the first 50 terms.

Nov 28, 2013 · Explore related questions sequences-and-series power-series See similar questions with these tags.

5 Find sum of series $$\sum_ {n=1}^ {\infty} \frac { (-1)^ {n+1}} {n^2}$$. I know the series converge absolutely so it is clearly convergent and in the absolute case the sum is $\pi^2/6$. …

Oct 22, 2018 · 3 If an alternating series converges, then adding the first $n$ terms of the series will approximate the final (infinite) sum with error at most the next term in the series.

Nov 4, 2024 · First observe that this is an alternating series and the magnitude of each term is decreasing. Hence by the Leibniz's test we conclude the series is convergent.