I was reading the definition of $\textit{Alternating Series}$, in wikipedia.
https://en.wikipedia.org/wiki/Alternating_series
All terms $a_n>0$ and the signs of the general terms alternate between positive and negative.
Now if we see the series $\Sigma_1^{\infty}(-1)^n \frac{\sin{n}}{n}$, the signs of general terms do not change signs alternatively. After 2nd term, two terms are negative and one term positive followed by two terms positive and one term negative.
How does this series fits with the definition of $\textit{Alternating Series}$? If not, then how can we use Leibniz Test to claim Conditional Convergence of this series?
Best Answer
Since it is not really an alternating series, you can't apply the alternating series test directly. However, you may apply the Dirichlet's test which is more general than the alternating series test.
Hint:
Let $a_n(x)=\frac1n$ and $b_n(x)=(-1)^{n}\sin(n)$ . Clearly, $a_n$ is monotonically decreasing and $\lim_{n\rightarrow \infty}a_n=0$.
Now show that $\bigg|\sum_{n=1}^{N}b_n\bigg|$ is bounded for every positive $N.$