Proving a series diverges

Question

Hello I am trying to prove the following series diverges $$\sum_{k=1}^{\infty} \ln\left(1+\frac{(-1)^{k+1}}{\sqrt{k+1}}\right)$$ This series alternates around 0 and goes to zero but fails the alternating test as it is not decreasing. Can someone give me a hint as to what test is appropriate here?

Answer 1

Since, by the Taylor expansion, as $x \to 0$, you have $$ \ln (1+x)=x-\frac12x^2+\mathcal{O}(x^3) $$ Then you may write, for $k \to +\infty$, $$ \ln\left(1+\frac{(-1)^{k+1}}{\sqrt{k+1}}\right)=\frac{(-1)^{k+1}}{\sqrt{k+1}}-\frac{1}{2(k+1)}+\mathcal{O}\left(\frac{1}{(k+1)^{3/2}}\right) $$ giving $$ \sum\ln\left(1+\frac{(-1)^{k+1}}{\sqrt{k+1}}\right)=\sum \frac{(-1)^{k+1}}{\sqrt{k+1}}-\sum \frac{1}{2(k+1)}+\sum\mathcal{O}\left(\frac{1}{(k+1)^{3/2}}\right), $$ on the right hand side, the first and the last series are convergent, the second (harmonic) series is divergent, thus your initial series is divergent.