How can i check if this sum is converges?

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The discussion centers on determining the convergence of a series, specifically whether the sum converges when using the variable $t$ instead of $w$. The series converges by comparison with the known convergent series $$\sum \frac{1}{t^2}$$ utilizing the limit comparison test. Additionally, the series can be expressed using partial fractions as $$\sum_{t=1}^\infty \left(\frac{-3/2}{t+3} + \frac{4}{t+4} + \frac{-5/2}{t+5}\right)$$, which results in a telescoping sum with a total of 5. However, if the variable is indeed $w$, the series diverges due to the uniformity of terms.

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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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