Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [telescopic-series]

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

In mathematics, a telescoping sum $(a_n)_n$ is a series whose general term $a_n$ can be decomposed as the difference between two consecutive terms of another series $(b_n)_n$, so that only a finite number of terms is left in the sum. $$\sum_{i = 1}^N a_n = \sum_{i = 1}^N (b_n - b_{n-1}) = b_N - b_0.$$ In particular, if $\lim\limits_{N \to \infty} b_N \to 0$, $$\sum_{i = 1}^\infty a_n = - b_0.$$

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How to prove that: $\tan(3\pi/11) + 4\sin(2\pi/11) = \sqrt{11}$

How can we prove the following trigonometric identity? $$\displaystyle \tan(3\pi/11) + 4\sin(2\pi/11) =\sqrt{11}$$
Parik
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Series involving Fibonacci Numbers: $\sum_{k=1}^\infty \frac{1}{F_kF_{k+1}}$

I will start my question with a bit of information that I think may be helpful to potential answerers. If you don't want to read it, skip down to the question. BACKGROUND: I'm investigating series in the form $$\Phi_n(x):=\sum_{k=1}^\infty…
Franklin Pezzuti Dyer
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Writing integers as a product of as few elements of $\{\frac21, \frac32, \frac43, \frac54, \ldots\}$ as possible

This question is inspired by question 2 of the 2018 European Girls' Mathematical Olympiad. Also posted on mathoverflow. Any integer $x \ge 2$ can be written as a product of (not necessarily distinct) elements of the set $A = \{\frac21, \frac32,…
user133281
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When is the series $\sum_{n=1}^\infty \frac1{a n^2 + b n + c}$ rational?

Let $a,b,c$ be integers such that $a\neq 0$ and $$ a n^2 + b n + c \neq 0 $$ for all positive integers $n$. (a) Prove that if there exists a positive integer $k$ such that $$ b^2 - 4ac = k^2 a^2, $$ then $$ \sum_{n=1}^\infty \frac{1}{a n^2 + b n +…
Frank
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Find the sum $\frac{1}{\sqrt{1}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + ...+ \frac{1}{\sqrt{99}+\sqrt{100}}$

I would like to check I have this correct Find the sum $$\frac{1}{\sqrt{1}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + ...+ \frac{1}{\sqrt{99}+\sqrt{100}}$$ Hint: rationalise the denominators to get a 'telescoping' sum: a sum of terms in which…
mikoyan
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The sum of series with natural logarithm: $\sum_{n=1}^\infty \ln\left(\frac{n(n+2)}{(n+1)^2}\right)$

Calculate the sum of series: $$\sum_{n=1}^\infty \ln\left(\frac{n(n+2)}{(n+1)^2}\right)$$ I tried to spread this logarithm, but I'm not seeing any method for this exercise.
Yas
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Proving $\sum_{n=1}^{99}\frac{\sqrt{n+1}-\sqrt{n}}{2n+1}\lt\frac9{20}$

I found the original question asked by someone else, asking for this to be proven using only '9th grade math', this is the image: Which can be written like $$\sum_{n=1}^{99}\frac{\sqrt{n+1}-\sqrt{n}}{2n+1}\lt\frac9{20}$$ Rationalizing it, I…
Typo
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The number of ways to represent a natural number as the sum of three different natural numbers

Prove that the number of ways to represent a natural number $n$ as the sum of three different natural numbers is equal to $$\left[\frac{n^2-6n+12}{12}\right].$$ It was in our meeting a year ago, but I forgot, how I proved it. Let the needed number…
Michael Rozenberg
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Different ways to come up with $1+2+3+\cdots +n=\frac{n(n+1)}{2}$

I am trying to compile a list of the different ways to come up with the closed form for the sum in the title. So far I have the famous Gauss story, the argument by counting doubletons, and using generating functions. Is there any other…
Ittay Weiss
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Alternative way to solve a limit problem

$$ \lim _{n \rightarrow \infty} \frac{1}{1+n^{2}}+\frac{2}{2+n^{2}}+\cdots+\frac{n}{n+n^{2}} $$ I want to find the limit of this infinite series which I found in a book. The answer is $1/2$. The solution to this limit was given by Sandwich/Squeeze…
batchcoding____s
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When does the limit of $a_n$ exist where $a_{n+1}:=a_n+\frac{a_n^2}{n^2}?$

Consider the recursive relation $a_{n+1}:=a_n+\cfrac{a_n^2}{n^2}$. The existence of $\lim_n a_n$ depends on the initial value $a_1$. For instance: If $a_1=1$, then $a_n=n$ and the sequence is divergent. If $a_1=0$, then $a_n=0$ and the sequence is…
Aster
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Evaluating $\sum_{r=1}^{\infty} \cot^{-1}(ar^2+br+c)$

Evaluate the series $$S=\sum_{r=1}^{\infty} \cot^{-1}(ar^2+br+c)$$ I have tried many values of $(a,b,c)$ and plugged into Wolframalpha, it always converges. I know that for particular values of $a,b,c$, we solve it by forming a telescoping series…
V.G
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Why does $\sum_1^\infty \frac1{n^3}=\frac52\sum_1^\infty\frac{(-1)^{n-1}}{n^3\binom{2n}{n}}$?

Apery's original proof that $$ \zeta(3) \equiv \sum_1^\infty \frac1{n^3} $$ is irrational starts from an alternating series $$ \zeta(3) = \frac52\sum_1^\infty\frac{(-1)^{n-1}}{n^3\binom{2n}{n}} $$ There must be a way to see that those two series…
Mark Fischler
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Prove $\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + \cdots$ converges to $\frac 1 2 $

Show that $$\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + \cdots = \frac{1}{2}.$$ I'm not exactly sure what to do here, it seems awfully similar to Zeno's paradox. If the series continues infinitely then each term is just going…
Jack Thompson
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Is every series a telescoping series?

This question may seem silly at first. We say that a series $\sum a_n$ is a telescoping series if there exists a sequence $(b_n)$ with $a_n=b_n-b_{n+1}$ for every $n$. One can show that $\sum a_n$ converges if and only if the sequence $(b_n)$…
User
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