Since the limit $\lim_{n\to\infty}(-1^n)\frac n{n+5}$ does not exist, then, in particular, you don't have $\lim_{n\to\infty}(-1^n)\frac n{n+5}=0$, and therefore the series diverges.
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Since the limit $\lim_{n\to\infty}(-1^n)\frac n{n+5}$ does not exist, then, in particular, you don't have $\lim_{n\to\infty}(-1^n)\frac n{n+5}=0$, and therefore the series diverges.