Conditional convergence $\sum_{k=1}^{\infty}\sin(\pi(k+\frac{1}{k}))$ and alternating series test

Question

I know that the following series

$$\sum_{k=1}^{\infty} \sin\left(\pi\big(k+\frac{1}{k}\big)\right)$$

is alternating and that it converges, but I have a question regarding the alternating series test.

The test basically says that for an alternating series, the series converges if $a_k \rightarrow 0$ as $k \rightarrow \infty$ and $a_k$ is monotonically decreasing.

Normally, an alternating series would be written in the form

$$\sum_{k=1}^{\infty} (-1)^k a_k $$

or something similar to that. But in the case with $\sum_{k=1}^{\infty} sin(\pi(k+\frac{1}{k})$, I simply don't know which part of it $a_k$ is. Any advice regarding this?

Answer 1

You may observe that $$ \sin\left(\pi\big(k+\frac{1}{k}\big)\right)=\sin(\pi k)\cos\left(\frac{\pi}{k}\right)+\sin\left(\frac{\pi}{k}\right)\cos(\pi k)=0+(-1)^k\sin(\frac{\pi}{k}) $$ using, for $k=1,2,3,\cdots,$ $$ \sin(\pi k)=0, \qquad \cos(\pi k)=(-1)^k. $$