# Thread: Absolute convergence

1. ## Absolute convergence

Does the sequence $\sum_{n=1}^\infty \frac{sin(2n)}{1+n+n^2}$ converge? If it does, does it converge absolutely?
I managed to get this series to converge but I have no idea whether it will converge absolutely.

I gather that there will no longer be any positive terms but I can't prove whether it is absolutely convergent or not.

Help would be appreciated!

2. Hello,
Originally Posted by Showcase_22
I managed to get this series to converge but I have no idea whether it will converge absolutely.

I gather that there will no longer be any positive terms but I can't prove whether it is absolutely convergent or not.

Help would be appreciated!
I trust you for the convergence part.

Now for the absolute convergence part, you have to prove that $\sum \left|\frac{\sin(n)}{1+n+n^2}\right|$ converges.
And we know that $|\sin(n)| \leq 1$, $1+n+n^2 \geq n^2$

Hence $\left|\frac{\sin(n)}{1+n+n^2}\right|\leq \frac{1}{n^2}$

So $\sum_{n \ge 1} \left|\frac{\sin(n)}{1+n+n^2}\right|\leq \sum_{n \ge 1} \frac{1}{n^2}$
And since the RHS is a convergence series, the LHS...

3. Originally Posted by Moo
Hello,

I trust you for the convergence part.
lol, well that's good! We are officially friends on here after all......

Before I forget: Merry Christmas!!

Now for the absolute convergence part, you have to prove that $\sum \left|\frac{\sin(n)}{1+n+n^2}\right|$ converges.
And we know that $|\sin(n)| \leq 1$, $1+n+n^2 \geq n^2$

Hence $\left|\frac{\sin(n)}{1+n+n^2}\right|\leq \frac{1}{n^2}$

So $\sum_{n \ge 1} \left|\frac{\sin(n)}{1+n+n^2}\right|\leq \sum_{n \ge 1} \frac{1}{n^2}$
And since the RHS is a convergence series, the LHS...
hmm, you made that look so easy!

I was getting confused that $1 \leq sin (n) \leq 1$ and it would have been a simple step to $|sin (n)| \leq 1$. I'm not sure what I was thinking about.