Given that $\sum\limits_{n=1}^{\infty}a_n$ converges ($a_n >0$), then does $\sum\limits_{n=1}^{\infty}a_n^{3} \sin(n)$ converge?

Given that $$\displaystyle \sum_{n=1}^{\infty}a_n$$ converges ($$a_n >0$$), then $$\displaystyle \sum_{n=1}^{\infty}a_n^{3} \sin(n)$$:

a) converges

b) diverges

c) does not exist

d) None of the above

My attempt: If we were given that $$(a_n)$$ is monotonically decreasing, then using the Abel-Olivier-Pringsheim criterion (Theorem 2.16 in these notes) we could conclude that $$\displaystyle \lim_{n\to \infty} na_n=0$$ from which we obtain $$k \in \mathbb{N}$$ such that for all $$n \geq k,$$ we have $$na_n<1$$ which implies $$|a_n^{3}\sin(n)|<\dfrac{1}{n^3}$$ which by the comparison test means that $$\displaystyle \sum_{n=1}^{\infty}a_n^{3} \sin(n)$$ converges absolutely.

But we are not given that $$(a_n)$$ is monotonically decreasing. At best, we have a monotonically decreasing subsequence tending to $$0.$$ Any hints on how to proceed?

• Hint: $a_n\to0$, hence there exists $k$ with $|a_n|\le k$ for all $n$. Dec 8, 2017 at 16:51

Since $\sum_n a_n$ converges, $a_n\to 0$. There is some $N$ such that $n\geq N\implies 0<a_n\leq 1$.

For $n\geq N$, $|a_n^{3} \sin(n)|\leq a_n^3\leq a_n$.

Therefore, $\sum_n a_n^{3} \sin(n)$ converges absolutely, hence converges.

• Thanks, was definitely overthinking it. Dec 8, 2017 at 17:07

Eventually, the $a_n$s are small. Then $$|a_n^3 \sin n| \leq |a_n|^3 \leq |a_n| = a_n.$$ Now compare and use absolute convergence.

If $$\sum_{n=1}^{\infty} a_n$$ converges, then $$a_n \to 0$$ and thus you can find $$N$$ sufficiently large enough so that $$a_n <1$$ for $$n > N$$. After this $$a_n^3 < a_n$$ for all $$n > N$$. This tell us that

$$\sum_{n=1}^{\infty} a_n^3 = \sum_{n=1}^{N} a_n^3 + \sum_{n=N+1}^{\infty} a_n^3 < \sum_{n=1}^{N} a_n^3 + \sum_{n=N+1}^{\infty} a_n <\infty$$

And finally we have $$\sum_{n=1}^{\infty} a_n^3 \sin(n) < \sum_{n=1}^{\infty} a_n^3$$

since $$\sin(n) < 1$$.