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The alternate series $S=\displaystyle \sum_{k=2}^{\infty} \frac{(-1)^n}{\sqrt{n}+(-1)^n} $ converges?

$S$ is absolutely convergent?

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marked as duplicate by David Mitra, Martin Sleziak, Daniel Fischer, Tomás, Gina Jul 21 at 14:30

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sorry to ask this, but what is the idea behind converging? I am hearing a lot on this site.. –  MonK Jul 21 at 13:31
    
have you done any work here? –  5xum Jul 21 at 13:31
1  
@Sid Converging is when a series, function, etc. gets closer to a specific value. Like how 0.9, 0.99, 0.999... gets closer to 1. –  Eul Can Jul 21 at 13:33
    
y tried using leibniz rule but not works $a_n=\frac{1}{\sqrt(n)+(-1)^n}$ is not a monotonic sequence and not converges to 0 and I don't know calculate $\lim an$ –  Pablo Herrera Jul 21 at 13:37
    
So in this question, S is getting or will get closer to 1? (a specific arbitrary value) –  MonK Jul 21 at 13:38

1 Answer 1

We have using Taylor series

$$ \frac{(-1)^n}{\sqrt{n}+(-1)^n}=\frac{(-1)^n}{\sqrt n}\left(1+\frac{(-1)^n}{\sqrt n}\right)^{-1}=\frac{(-1)^n}{\sqrt n}-\frac1n+\mathcal O\left(\frac1{n^{3/2}}\right)$$ so we see that the given series is the sum of $3$ series: two convergent series using Leibniz theorem and using the comparison with a Riemann series and a harmonic series which is divergent. Hence the given series is divergent.

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taylor series seriously? it never be in my plans hahaha thanks a lot –  Pablo Herrera Jul 21 at 13:44
    
You're welcome. –  Sami Ben Romdhane Jul 21 at 13:45

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