How can i check if this sum is converges?

[DrMath1]

Hello everybody,
How can i check if this sum is converges?
View attachment 6649

 

[Opalg]

Hi Dr Math, and welcome to MHB!

I assume that the symbol under the summation sign should be $t$ rather than $w$? If so, then the series converges, by comparison with the known convergent series \(\displaystyle \sum \frac1{t^2}\) (using the limit comparison test).

If you want, you can use partial fractions to write the series as \(\displaystyle \sum_{t=1}^\infty \left(\frac{-3/2}{t+3} + \frac4{t+4} + \frac{-5/2}{t+5}\right).\) This is a telescoping sum, with sum $5$.

Edit. On second thoughts, I'm not so sure about the $t$ and $w$. There is a $t$ on the left side of the equation, which goes against the assumption that $t$ is the summation index. If the summation is really over another variable $w$, then each term in the sum is the same (because there are no $w$s in it). That means that the series will diverge.