Interchanging a limit and an infinite alternate series

I am having troubles to explain if the following equality holds or not
$$lim_ktoinftysum_n=1^infty(-1)^n(n+k)^-1=sum_n=1^infty(-1)^nlim_ktoinfty(n+k)^-1=0.$$
As far as I see, I can't apply the dominated convergence theorem since $|f_k(n)|=|(-1)^n(n+k)^-1|=(n+k)^-1$ can't be dominated by summable sequence over $n$. How could I proceed?

2 Answers
2

HINT:

Note that we can write

$$sum_n=1^infty frac(-1)^nn+k=sum_n=1^inftyleft(frac12n+k-frac12n-1+kright)$$

Can you proceed now?

Or:
$$ sum_ngeq 1frac(-1)^nn+k = sum_ngeq 1int_0^+infty(-1)^n e^-nx e^-kx,dx =int_0^+inftyfracdx(1+e^x)e^kxleq int_0^+inftyfracdx2e^kx=frac12k.$$

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