# Questions tagged [telescopic-series]

For summation questions involving telescopic sums/series. This tag is often used with (summation) or (sequences-and-series).

146 questions
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### how to write the general formula for a telescoping series

Im trying to check if the series $\sum_{k=2}^{\infty} \frac{1}{\sqrt{k-1}} - \frac{1}{\sqrt{k+1}}$ is converging or not. I have divided the series into two parts with the series with even k >= 2 ...
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### Sum of a Sequence of Odd Numbers that are Squared [duplicate]

What is the sum of all the numbers in the sequence $1^2 + 3^2 + 5^2 + 7^2 + 9^2 + \ldots + k^2$. Note that all the numbers being squared in the sequence are all odd numbers. This is what I have done ...
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### Rigorously proving that $\sum_{k=1}^{n}\sum_{j=1}^{k}d_j$ is equal to $\sum_{k=0}^{n-1}(n-k)d_{k+1}$

The way that I originally arrived upon $\sum_{k=0}^{n-1}(n-k)d_{k+1}$ is by seeing that $\sum_{k=1}^{n}\sum_{j=1}^{k}d_j$ is similar to a telescoping sum, as when the sums for various values of $k$ ...
3answers
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### Help summing the telescoping series $\sum_{n=2}^{\infty}\frac{1}{n^3-n}$.

I know a priori that the series $$\sum_{n=2}^{\infty}\frac{1}{n^3-n}$$ converges. However, I am tasked with summing the series by treating it as a telescoping series. By partial fraction ...
1answer
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### A Peculiar Sum Of Squares [duplicate]

I have been trying to solve a question that arose in my mind a couple of days ago. What is the sum of: $1+4+9+16+25+\cdots +n^2$? It is the sum of squares of each numbers starting from $1$ to $n$....
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