# How do you determine the convergence or divergence of #Sigma ((-1)^(n)n)/(n^2+1)# from #[1,oo)#?

As:

and:

the series is convergent.

An alternative series:

So we make the test:

As the denominator is always positive we can focus on the numerator:

#( (n+1)(n^2+1)-n( (n+1)^2+1)) = n^3+n+n^2+1 -n (n^2+2n+1+1) = n^3+n+n^2+1 -n^3-2n^2-2n = -2n^2-2n +1 <=0#

The we check that:

So both conditions are satisified and the series is convergent.

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Using this in conjunction with the fact that

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