MHB How can i check if this sum is converges?

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The discussion centers on determining the convergence of a series. Initially, it is suggested that the summation index should be $t$ instead of $w$, leading to the conclusion that the series converges by comparison with the convergent series $$\sum \frac{1}{t^2}$$ using the limit comparison test. However, upon further reflection, the possibility of $w$ being the correct index raises concerns, as it would result in each term being identical, causing the series to diverge. The conversation highlights the importance of correctly identifying the summation variable to assess convergence accurately. Ultimately, the convergence of the series depends on the proper interpretation of the summation index.
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Hello everybody,
How can i check if this sum is converges?
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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 $$\sum \frac1{t^2}$$ (using the limit comparison test).

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