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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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66 views

sum of the series: $\cos^3 \alpha +\cos^3 {3\alpha} + \cos^3 {5\alpha}+…+\cos^3 {(2n-1)\alpha}$. [duplicate]

Question: Find the sum of the series: $$\cos^3 \alpha +\cos^3 {3\alpha} + \cos^3 {5\alpha}+\ldots+\cos^3 {(2n-1)\alpha}$$ The book from which this question was taken says that the answer is $$\...
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1answer
60 views

How to compute the $n^{th}$ partial sum of a series?

Compute explicitly $S_n(x)$, the $n^{th}$ partial sum of the series $$\sum_{k=1}^∞ \frac{x\left[-1+4k(k+1)x^2\right]}{(1+4k^2x^2)(1+4(k+1)^2x^2)}$$ then compute the sum $S(x)$ of the infnite series, ...
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0answers
28 views

Computing $S_n(x)$, the partial sum of a series explicitly

Compute explicitly $S_n(x)$, the $n^{th}$ partial sum of the series $$\sum_{k=1}^∞ \frac{x\left[-1+4k(k+1)x^2\right]}{(1+4k^2x^2)(1+4(k+1)^2x^2)}$$ then compute the sum $S(x)$ of the infnite series, ...
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3answers
79 views

Find the sum of the series: $\cos^3 \alpha +\cos^3 {3\alpha} + \cos^3 {5\alpha}+…+\cos^3 {(2n-1)\alpha}$.

Question: Find the sum of the series: $$\cos^3 \alpha +\cos^3 {3\alpha} + \cos^3 {5\alpha}+....+\cos^3 {(2n-1)\alpha}$$ The book from which this question was taken says that the answer is $\frac{3\...
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4answers
55 views

Could I get an explanation for the telescoping series? [closed]

I am confused on how to solve a $S_N$ using a telescoping series. I don't know where the $\frac{1}{N+1}$ comes from. The problem reads, Let S = $\sum\limits_{n=1}^\infty (\frac{1}{n} - \frac{1}{n+...
2
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3answers
56 views

Find the sum: $\sum_{n=1}^{\infty} \ln\left(\frac{b+n+1}{a+n+1}\right)$

Let $0<a<b$, I would like to compute the sum $$\sum_{n=1}^{\infty} \ln\left(\frac{b+n+1}{a+n+1}\right).$$ But first I am worrying that a test convergence might lead to the divergence of this ...
2
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2answers
37 views

$\sum_{n=q_1}^{q_2-1}\frac{1}{n^2}<\frac{1}{q_1-1}$ inequality

How can We prove that $$\sum_{n=q_1}^{q_2-1}\frac{1}{n^2}<\frac{1}{q_1-1}$$ for integer $q_1,q_2$ : $1\le q_1 \le q_2 $. Obvious esimation gives $$\sum_{n=q_1}^{q_2-1}\frac{1}{n^2}<\frac{q_2-q_1}...
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2answers
203 views

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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3answers
57 views

Approximating series of fractions [duplicate]

Let $$ P = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}}+ \frac{1}{\sqrt{4}} ... +\frac{1}{\sqrt{10000}}$$ what is the value of the floor function of P? My try: I tried assuming these 2 bounds $$ P_x =...
1
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1answer
40 views

Sums with gcd $\sum_{n=1}^{1009} \gcd(2n, 2020) - \sum_{n=0}^{1008} \gcd(2n+1, 2019)$

The problem $\sum\limits_{n=1}^{1009} \gcd(2n, 2020) - \sum\limits_{n=0}^{1008} \gcd(2n+1, 2019)$, has stumped me for a while. I plugged it into Wolfram Alpha, and got a 4-digit number. However, I ...
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3answers
52 views

Trigonometric series sum with $\sin$ function

Sum of Trigonometric series $\sin (x)-\sin(2x)+\sin(3x)-\sin(4x)+\cdots n$ terms Try: I have take $2$ cases $\bullet$ If $n$ is even natural number, Then $S=\sin(x)-\sin(2x)+\sin(3x)+\cdots -...
2
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2answers
72 views

Question from the 2011 IMC (International Mathematics Competition) Key Stage III paper, about the evaluation of a quadratic equation

When $a=1, 2, 3, ..., 2010, 2011$, the roots of the equation $x^2-2x-a^2-a=0$ are $(a_1, b_1), (a_2, b_2), (a_3, b_3),\cdots, (a_{2010}, b_{2010}), (a_{2011}, b_{2011})$ respectively. Evaluate: $...
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1answer
53 views

Evaluating $S_n$ where $n=1,2,3,\dots$ and $S_n=\sum_{k=1}^{\infty }\frac{1}{(4k^2-1)^n}$

If $n$ is a natural number and $S_n=\sum_{k=1}^{\infty }\frac{1}{(4k^2-1)^n}$, then $S_1=\frac{1}{2}, S_2=\frac{\pi^2-8}{16}, S_3=\frac{-3\pi^2+32}{64}, S_4=\frac{\pi^4+30\pi^2-384}{768},\dots$. How ...
3
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4answers
92 views

Compute $\sum_{k=1}^{25} (\frac{1}{k}-\frac{1}{k+4})$

Compute $\sum_{k=1}^{25} (\frac{1}{k}-\frac{1}{k+4})$ I know that some of the terms will cancel each other. Have it been $k+1$ instead of $k+4$, I could have easily see the pattern in which the terms ...
6
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2answers
207 views

$\sum\limits_{n=1}^{\infty}\arctan{\frac{2}{n^2+n+4}}$

$$\sum\limits_{n=1}^{\infty}\arctan{\frac{2}{n^2+n+4}}$$ We know that : $\arctan{x} - \arctan{y} = \arctan{\frac{x-y}{1+xy}}$ for every $ xy > 1 $ I need to find two numbers which satisfy: $ab = ...
2
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1answer
42 views

Sum of series: $1*3*(2^2) + 2*4*(3^2) + 3*5*(4^2) + \dots$?

I am trying to find the sum of the above series. The sum till n terms can be found using power series expansion. However, I'm trying to solve this using the method of difference (a.k.a. Telescoping ...
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6answers
271 views

Finding sum of none arithmetic series

I have a question to find the sum of the following sum: $$ S = \small{1*1+2*3+3*5+4*7+...+100*199} $$ I figured out that for each element in this series the following holds: $$ a_n = a_{n-1} + 4n - 3 $...
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2answers
173 views

How to solve $V(n) = 2 \cdot V(n-1) + 2 \cdot n$

How to solve $V(n) = 2 \cdot V(n-1) + 2\cdot n$? I've tried using telescoping, but I'm not able to get correct solution. The textbook has a solution with homogeneous and particular solution and then ...
1
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2answers
59 views

A Question based on series.

Find the sum of$$\sum _{r=1}^\infty\left({\frac{2}{4r-3}-\frac{1}{2r}}\right)$$ I tried to solve this problem by taking $2$ common from the expression and got the result as of$$2\sum _{r=1}^\infty\...
1
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1answer
77 views

Closed form for sum

I wonder whether there is a closed form for this sum $$ S_n:=\displaystyle\sum_{k=0}^n \dfrac{4^k}{4^k+5^k}$$ The question asks to express the sum in terms of $n$ then to deduce the limit of $\dfrac{...
0
votes
1answer
40 views

Sum of this converging telescoping series?

I'm trying to understand which is the sum of the following telescoping series (I showed this is converging, I'm not reporting here): $$\sum\limits_{j=n}^{\infty} [\mathbb{P}(E_j) - \mathbb{P}(E_{j+1})...
0
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2answers
41 views

How to prove this series is a telescoping series and calculate its addition [duplicate]

In class, they have told us that we need to prove that the following series is a telescopic one by using $ln$ features and to calculate its addition. It´s the first time we work with them, so I don´...
2
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1answer
70 views

Telescoping $\sum_{n\ge2} \ln(1-\frac1{n^2})$ leads to wrong result

I know that this series converges to $-\ln2$. I'm not looking for someone to show me that here. I've used a way of telescoping the series that gives a result of 0. I want to know what's wrong about my ...
0
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1answer
70 views

Solving recurrence using telescopic sums

Use the method of telescoping sums to solve the recurrence $nx_n = (n−2)x_{n−1} +1, n\geqq1$, where $x_0=0$. I did this using the method of inspection. In that I found that $x_n=\frac{1}{2}$ for all $...
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4answers
543 views

find the sum to n term of $\frac{1}{1\cdot2\cdot3} + \frac{3}{2\cdot3\cdot4} + \frac{5}{3\cdot4\cdot5} + \frac{7}{4\cdot5\cdot6 } + … $

$$\frac{1}{1\cdot2\cdot3} + \frac{3}{2\cdot3\cdot4} + \frac{5}{3\cdot4\cdot5} + \frac{7}{4\cdot5\cdot6 } + ... $$ $$=\sum \limits_{k=1}^{n} \frac{2k-1}{k\cdot(k+1)\cdot(k+2)}$$ $$= \sum \limits_{k=...
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2answers
109 views

Sum of infinite telescoping series $\sum_{r=2}^\infty \frac{1}{r^2-1}$? [duplicate]

How do I find sum of $\sum_{r=2}^\infty \frac{1}{r^2-1}$? The answer given in my book is 3/4. I can decompose the general term into $(\frac{1}{r-1}-\frac{1}{r+1})$ multiplied by 1/2 but since it is ...
0
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1answer
25 views

Prove $x_n = \sum_{k=1}^n \frac{1}{(a+(k-1)\cdot d)\cdot(a+k\cdot d)}$ is a bounded sequence.

Let $n \in \mathbb N$ and: $$ x_n = \sum_{k=1}^n \frac{1}{(a+(k-1)\cdot d)\cdot(a+k\cdot d)} $$ Prove $\{x_n\}$ is a bounded sequence. I'm having hard time finishing the proof. Below is what i've ...
2
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3answers
156 views

Prove $\sum_{1}^{\infty} \frac{1}{\sqrt{n}( n + \sqrt{n})} \lt 2 $

Prove that $$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}( n + \sqrt{n})} \lt 2$$ I have found that $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}( n + \sqrt{n})} < \pi / 2 $ with integrating from $1$ to ...
2
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2answers
68 views

Generalized Telescoping Series

Goog Morning Everyone, I'd like to ask the following exercise that my professor gave, that i think it has more theory behind it than expected it. The exercise ask to prove what the following ...
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1answer
83 views

Finding the nth partial sum of a telescoping series

$\sum _{n=1}^{\infty }\:\left[\frac{2}{\left(n+1\right)}-\frac{2}{\left(n+3\right)}\right]$ This question is off from webwork and I already got everything right except for finding the nth partial sum....
2
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4answers
98 views

Computing the value of $\frac{1}{3^2+1} + \frac{1}{4^2+2} + \frac{1}{5^2+3}…\infty$=?

I have tried converting this series into a telescopic sum whose terms could cancel out but haven't succeeded in that effort. How should I proceed further?
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4answers
348 views

Prove Summation $k=1$ to $n$ $k^3$ with telescoping rule

I know how to do this problem when trying to get sum of squares $$\Sigma k^2 = n(n+1)(2n+1)/ 6 $$ But I’m having trouble proving for cubes: $$\sum_{k=1}^n k^3 = \frac{n^2(n+1)^2}4$$ I have to ...
3
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4answers
66 views

Show that the series converges and find its sum [duplicate]

Show that $$ \sum_{n=1}^\infty \left( \frac{1}{n(n+1)} \right) = \frac{1}{2}+\frac{1}{6}+\frac{1}{12}+ \;... $$ converges and find its sum. My solution so far: I am thinking about finding the ...
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4answers
65 views

$n$-th partial sum and convergence $\sum_{k=1}^{\infty}\frac{1}{k(k+2)}$

Having trouble with finding the $n$-th partial sum, and seeing if it diverges or not of, $$\sum_{k=1}^{\infty}\frac{1}{k(k+2)}$$ I know that it is a telescoping series, and I can solve $\...
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1answer
60 views

What is the value of n such that n! = 3! × 5! × 7!

does this use the telescopic series if it does how do you express the general term? I have no idea. I need some help
0
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1answer
49 views

Find the largest prime factor of $1+f(1)+f(2)+f(3)+\dots+f(30)$ where $f(n)=n\cdot n!$

Here's my approach: express the sum as a telescoping series although I am not sure how to go about it I am sure it is either 29 or 31. Could someone help me?
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4answers
98 views

Sum of n terms of this series

$\frac{1}{1.3} + \frac{2}{1.3.5} +\frac {3}{1.3.5.7} + \frac{4}{1.3.5.7.9}........ n $ Terms. I Know the answer to this problem but I couldn't find any proper way to actually solve this question. ...
8
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4answers
159 views

Prove that inequality $1+\frac{1}{2\sqrt{2}}+…+\frac{1}{n\sqrt{n}}<2\sqrt{2}$

Let $n$ is a natural number. Prove that inequality $$1+\frac{1}{2\sqrt{2}}+\frac{1}{3\sqrt{3}}+...+\frac{1}{n\sqrt{n}}<2\sqrt{2}$$ My try: $$\frac{1}{n\sqrt{n}}=\frac{\sqrt{n}}{n^2}<\frac{\sqrt{...
3
votes
3answers
297 views

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 ...
2
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2answers
54 views

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 $k$ ...
5
votes
3answers
332 views

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 ...
0
votes
1answer
54 views

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 $n$....
1
vote
0answers
27 views

Interesting Recursive (Telescoping?) Inequality [duplicate]

Let $x_0 = 5,$ and $x_{n+1} = x_n + \frac {1} {x_n}$ for n = $0, 1, 2, . . ..$ Show that $45 < x_{1000} < 45.1.$ First noticing that $x_{n+1}-x_n= \frac {1}{x_n},$ we get $\sum \limits_{n=1}^{...
2
votes
2answers
122 views

Find $x$ if $\frac1{\sin1°\sin2°}+\frac1{\sin2°\sin3°}+\cdots+\frac1{\sin89°\sin90°} = \cot x\cdot\csc x$ [duplicate]

If $$\dfrac1{\sin1°\sin2°}+\dfrac1{\sin2°\sin3°}+\cdots+\dfrac1{\sin89°\sin90°} = \cot x\cdot\csc x$$ and $x\in(0°,90°)$, find $x$. I tried writing in $\sec$ form but nothing clicked. Any ideas?
0
votes
3answers
81 views

Finding $\Sigma\frac{(n+1)(n+2)}{2!}$ .

How to find the summation of $\Sigma\frac{(n+1)(n+2)}{2!}$ ? MY WORK: I know that the expression in the summation is the general term of the binomial expansion $(1-x)^{-3}$ . I have a solution ...
0
votes
3answers
69 views

Simplify $\sum_{k=0}^{n} \frac{1}{(n-k)!(n+k)!}$ [closed]

I have to simplify the following sum ($n \in \mathbb{N}$). I have tried to give values to $n$, but I haven't noticed anything useful. Can you help me, please? Thanks! $$\sum_{k=0}^{n} \frac{1}{(n-k)!(...
1
vote
1answer
38 views

Which partial sum should be considered to find $\sum_{n=1}^\infty c_1x_{n+1}+c_2x_{n+2}+…+c_kx_{n+k}$

Let $x_n\to x$ and $k\in \mathbb N$. Consider the infinite series $\sum_{n=0}^\infty a_n$ where $a_n=x_n-x_{n+k}$. Then the partial sum $s_{mk-1}$ is equal to $$s_{mk-1}=x_0+x_1+...+x_{k-1}-(x_{mk}+...
6
votes
3answers
87 views

Find the value of $S$ in term of $k$ (telescoping sums)

Let $k=\frac{1}{1\times2}+\frac{1}{3\times4}+\frac{1}{5\times6}+\cdots+\frac{1}{2549\times2550}$. Find the value of $S=\frac{1275}{1276}+\frac{1276}{1277}+\frac{1277}{1278}+\cdots+\frac{2548}{2549}$ ...
2
votes
2answers
61 views

Finding $\frac1{1+1^2+1^4}+\frac2{1+2^2+2^4}+\cdots+\frac n{1+n^2+n^4}$. [duplicate]

Find an expression for $$\frac1{1+1^2+1^4}+\frac2{1+2^2+2^4}+\cdots+\frac n{1+n^2+n^4}.$$ This was given in the chapter for APs. However, I do not see how this relates to them. I tried using ...
0
votes
1answer
50 views

Find the value of the sum $\frac{1}{1000*1998}+\frac{1}{1001*1997}+\cdots+\frac{1}{1998*1000}$

Find the value of the sum $\frac{1}{1000*1998}+\frac{1}{1001*1997}+\cdots+\frac{1}{1998*1000}$ I attempted expressing it as a telescoping sum, but I don't know how to. Also, I was curious, is there a ...