Does this sum converge, is my solution good?

Question

$$ \sum_{n=1}^\infty \frac{\sin(n)^{7}}{(n^{7}+1)^{1/2}} $$

I would say that it doesn't converge, cause I would write this as: $$ $$

$$ \frac{\sin(n)^{7}}{(n^{7})^{1/2}} $$ when $$ \lim_{n\to \infty} $$ then I would write this as:

$$ \sum_{n=1}^\infty \frac{\sin(n)^{7}}{(n^{7})*n^{1/2}}$$ and then I would say that $$ \sum_{n=1}^\infty \frac{\sin(n)^{7}}{(n^{7})^{1/2}} $$ converges , but that the $$ \sum_{n=1}^\infty \frac{1}{n^{1/2}} $$

doesn't converge, so overall sum doesn't converge.

Answer 1

You are not applying well the properties of the exponentials. Instead, you can write: $$ \sum_{n=1}^\infty \Bigg| \frac{\sin(n)^7}{(n^7)^{1/2}}\Bigg| \leq \sum_{n=1}^\infty \frac{1}{n^{7/2}} <\infty,$$ since $$ \sum_{n=1}^\infty \frac{1}{n^{\alpha}}<\infty $$ if and only if $\alpha>1$.

Answer 2

The sum converges because $\sin(n)\leq 1$, so \begin{equation} \sum_{n=1}^{\infty} \Bigl| \frac{\sin^7(n)}{(n^7+1)^{1/2}} \Bigr| \leq \sum_{n=1}^{\infty} \frac{1}{(n^7+1)^{1/2}} < \sum_{n=1}^{\infty} \frac{1}{(n^7)^{1/2}} = \sum_{n=1}^{\infty} \frac{1}{n^{7/2}} \end{equation} The last series converges, so the given series converges absolutely. Absolute convergence implies convergence.