# Questions tagged [telescopic-series]

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

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### Using Gosper's algorithm to obtain the WZ certificate of $\sum \binom{n}{k} = 2^n$

I'm not sure where my work is wrong, I'm not obtaining an answer, even though I know there should be one. In order to obtain the WZ proof certificate for the sum $$\sum_{k=0}^n \binom{n}{k} = 2^n$$ ...
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### What is the value of $a_1a_2\cdots a_{2019}$?

Let $a_1=\frac 34$ and for any $n\geq2$ $4a_n=4a_{n-1}+\frac {2n+1}{1^3+2^3+\cdots n^3 }$. What is the value of $a_1a_2\cdots a_{2019}$? I tried $1^3+2^3+\cdots +n^3=\frac {n^2(n+1)^2}{4}$ and I ...
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### Sum $\sum_{n=1}^{\infty} {n2^n\over(n+2)!}$?

$$\sum_{n=1}^{\infty} {n2^n\over(n+2)!}$$ The exercise mentions that this can be written as a telescopic series; I've been trying to write it in such a way but I'm stuck, can't seem to find one! Any ...
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### Find the sum of series $\sum_{k=1}^\infty \frac{1}{k^2+2k}$ [duplicate]

Find the sum of the series.$$\sum_{k=1}^\infty \frac{1}{k^2+2k}$$ Which technique should I use? I tried but I cannot find anything.
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### Find the limit of $\frac{T_n}{5n+4}$

Given that $U_0=0$, $U_{n+1}=\frac{U_n+3}{5-U_n}$ Find the limit of $U_n$ Set $T_n= \sum_{k=1}^n \frac1{U_k-3}$, find $\lim_{n\to+\infty}\frac{T_n}{5n+4}$ Approaches So for the first question, I ...
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### Explicit formula for $S_n=\sum_{r=1}^n\dfrac1{r(r+1)}$. [duplicate]

I was asked to find an explicit formula for $$S_n=\sum_{r=1}^n\dfrac1{r(r+1)}$$ and then go on to find the limit. I deduced that it would give $S_n=\frac1n-\frac1{n+1},$ however I was wrong and the ...
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### How would I evaluate this series using telescopic summation?

The series is $$\frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!} ... + \frac{n-1}{n!},$$ which I can write as the sum $$\sum_{i=2}^n \frac{i-1}{i!}$$ I will try to evaluate the partial sums to look for a ...
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### Why does this telescoping series end up with a 2 in the numerator?

I am confused about how to solve for this series: $$f(x) = c/((x+1) (x+3))$$ for $x = 1, 2, 3...$ solving for $c$. I see this is a telescoping series with a result of $1/(x+1) - 1/(x+3)$ but the ...
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### How to use telescoping series to find: $\sum_{r=1}^{n}\frac{1}{r+2}$ [closed]

I am a bit confused in this one, how do i do the required modification in this case?
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