Questions tagged [telescopic-series]

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

Filter by
Sorted by
Tagged with
2 votes
0 answers
59 views

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 $$\sum_{k=0}^n \binom{n}{k} = 2^n$$ ...
user avatar
  • 2,445
2 votes
1 answer
61 views

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 ...
user avatar
  • 604
0 votes
1 answer
21 views

Upper bound for Telescoping sum in gradient descent

I am studying a chapter in gradient descent . At some point we reach the sum in the left of the enequality and the writer says it's telescopic so this enequality holds: $\sum_{t=1}^T \Big( ||x_t - x^*|...
user avatar
0 votes
1 answer
55 views

Summation of alternating series:$\sum_{k=1}^{n} (-1)^{k-1}k$

Alternate summations $S_1=1-2+3-4+5-......+(2m-1)$ and $S_2=1-2+3-4+5-......-2m$ can be found as $\pm m$, respectively by arranging $$S_1=[1+2+3+4+5+.....+(2m-1)]-4[1+2+3+4+....+m]$$ We can get the ...
user avatar
  • 38k
1 vote
5 answers
130 views

Infinite sums of squares [closed]

$$\sum_{n=0}^{\infty} \frac {k^2(1-k)^2}{(n+k)^2(n+1-k)^2}$$ Here can anyone help me to solve this question,I can't think of any logic like telescopic, coefficient compare etc . It would be helpful if ...
user avatar
  • 45
0 votes
0 answers
73 views

Is it a challenge to evaluate the indefinite integral $\int \frac{\sin n x}{\sin x} d x$, where $n\in N?$

Noting that \begin{aligned}I_{k}-I_{k-2} &=\int \frac{\sin k x-\sin (k-2) x}{\sin x} d x \\&=2 \int \frac{\cos (k-1) x \sin x}{\sin x} d x \\&=2 \int \cos (k-1) x d x \\&=\frac{2}{k-1} ...
user avatar
  • 5,442
1 vote
0 answers
47 views

Need help in showing that the summation $\sum_{k=1}^{n} (a_{k+1}-a_{k}) = a_{n+1}-a_1$ [duplicate]

Given a sequence of real number $a_{1}$,$a_{2}$,...,$a_{n+1}$ show that $\sum_{k=1}^{n} (a_{k+1}-a_{k}) = a_{n+1}-a_1$ I am stuck on this problem we have been given by my lecturer. I don't have much ...
user avatar
  • 87
0 votes
0 answers
52 views

How to convert this into a telescopic sum [duplicate]

I'm trying to convert this series into a telescopic series but I'm stuck on making it into a telescopic form by factoring it, tried partial faction decomposing it but couldn't proceed any further ...
user avatar
11 votes
6 answers
1k views

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 ...
user avatar
1 vote
0 answers
51 views

Expressing $\sum\limits_{k=1}^{n-1}\frac{1}{k(k+1)}$ as $1 - \frac{1}{n}$

I was reading from INTRODUCTION TO ALGORITHM (THIRD EDITION) By Thomas H. Cormen and came across Telescoping series in the Appendix A (page 1148). And this was the definition: For any sequence $a_0,...
user avatar
  • 315
0 votes
1 answer
95 views

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 ...
user avatar
  • 101
-1 votes
1 answer
89 views

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.
user avatar
2 votes
1 answer
81 views

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 ...
user avatar
0 votes
1 answer
69 views

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 ...
user avatar
12 votes
1 answer
202 views

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}}{...
user avatar
  • 361
2 votes
5 answers
121 views

Value of $\sum_{n=0}^{1947}\frac{1}{2^n+\sqrt{2^{1947}}}$ $?$ [duplicate]

Find the value of $$\sum_{n=0}^{1947}\frac{1}{2^n+\sqrt{2^{1947}}}$$ MY APPROACH : I'm not able to telescope it . I tried to Rationalize it , but It was not possible . I've not attached any solution ...
user avatar
2 votes
1 answer
59 views

How to evaluate the following Product?

so I have the following product to evaluate : $$P_{n}=\prod_{k=0}^{n-1} u_{k}$$ Where : $u_k = e^{w_n}$ All what I know about $u_k$ is that $$ u_{n}=\frac{n+2}{n+1} $$ So we'll have the following : $$ ...
user avatar
-1 votes
3 answers
93 views

Summation of a finite sequence

This question is linked from my previous question: Summation of a sequence? Given the sequence: $$ a_n = 0.9^{n-1}a_1(1+d+d^2+...d^{n-1}) $$ and $a_1=100$ , $d= 1.5$ How to form an equation to find: $$...
user avatar
  • 101
2 votes
3 answers
91 views

How to find if a series is telescopic

The series $$\sum_{n=1}^\infty\left(\frac{4n+4}{3n+1}-\frac{4n}{3n-2}\right)$$ is telescopic and it converges to $-4+\dfrac43$. But if we get the equivalent expresion $$\sum_{n=1}^\infty\frac{-8}{9n^2-...
user avatar
  • 62.7k
3 votes
1 answer
88 views

Summation of powers of r and reciprocal of binomial coefficient

Evaluate the following sum: $$\sum_{r=1}^m\frac{(m+1)(r-1)(m^{r-1})}{r\binom{m}{r}}$$ where $\binom{m}{r}$ stands for ${}^mC_r$ I initially tried to change this into $$\frac{m+1}{m}\sum_{r=1}^m\frac{...
user avatar
1 vote
1 answer
72 views

Prove that, $\sum_{k=1}^{n-1} \frac{1}{\sqrt{a_k} + \sqrt{a_{k+1}}} = \frac{n-1}{\sqrt{a_1} + \sqrt{a_n}}.$

The sequence $(a_k)_{k \geqslant 1}$ is an AP. Prove that, for every $n \in \mathbb{Z^+}$, $$\sum_{k=1}^{n-1} \frac{1}{\sqrt{a_k} + \sqrt{a_{k+1}}} = \frac{n-1}{\sqrt{a_1} + \sqrt{a_n}}.$$ I ...
user avatar
  • 365
0 votes
1 answer
42 views

Simplification of a telescopic series

I've read this question on Math SE, but I cannot figure out the following: How to simplify $S$ to be $S=2-\frac{1}{a_{101}}$.
user avatar
  • 85
3 votes
0 answers
40 views

How to compute the limit of the infinite series $\sum_{n=1}^{\infty} \frac{1}{n(n+1)\ldots(n+m)}$, $m \in \mathbf{N}$ [duplicate]

For having some fun with infinite series, I am having a go at some of the exercises in Problems in Mathematical Analysis by Kaczor and Nowak. I would like to ask, ...
user avatar
  • 4,494
3 votes
1 answer
52 views

Find $\sum_{i=1}^n \frac{\sqrt{i+1}-\sqrt{i}}{\sqrt{i^2+i}}$

$$\sum_{i=1}^n \frac{\sqrt{i+1}-\sqrt{i}}{\sqrt{i^2+i}}$$ I have tried simplifying but I get $$\sum_{i=1}^n \frac{1}{\sqrt{i}}-\sum_{i=1}^n \frac{1}{\sqrt{i+1}}$$ which is subtraction of two ...
user avatar
0 votes
2 answers
86 views

Compute using the telescoping series $\sum_{n=1}^{\infty} \frac{n}{(n+1)(n+2)(n+3)} = \frac{1}{4}$

This question is similarly found here, and I have been trying to break it down using partial fraction however I cannot seem to get the final result as in the thread linked. I would very much ...
user avatar
0 votes
0 answers
51 views

Prove the following series converges and has the sum indicated using telescoping: $\sum^{\infty}_{n=1}\frac{2}{3^{n-1}}=3$

I'm trying to compute the following limit for convergence using the telescoping property, however, I'm having difficulties simplifying the equation so the property can be used: The following is the ...
user avatar
2 votes
1 answer
119 views

Visual proof of $\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + \frac{1}{4 \times 5} + ... = 1$

I am trying to find a visual way of proving that $$\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + \frac{1}{4 \times 5} + ... = 1$$ I am familiar with algebraic methods of proving ...
user avatar
3 votes
1 answer
111 views

Prove that $\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{F_kF_{k+1}}=\frac{1}{\phi}$.

In this question, the OP says that $$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{F_kF_{k+1}}=\frac{1}{\phi}$$ where $F_n$ is the $n$th Fibonacci number, defined by $F_n=F_{n-1}+F_{n-2}$, with $F_1=F_2=1$. I'...
user avatar
3 votes
3 answers
160 views

Does $\sum^{\infty}_{n=1} \frac{1}{a_n+1}$ with $a_{n+1}=a_n(a_n+1)$ and $a_1=\frac12$ have a closed form?

Mr Beast here. I've been working on a complicated math problem, and I obtained a recursive sequence $a_{n+1}=a_n(a_n+1)$ and that $a_1=\frac 12$. I've been thinking, is it possible to convert this to ...
user avatar
0 votes
1 answer
126 views

Finding a formula for the kth partial sum

I need help finding the kth partial sum of the question below: Consider the series $\sum_{n=1}^{\infty}a_n$ where $a_n=\ln\left(\frac{6n-1}{6n+5}\right)$. Find a formula, in closed form, for the $k^{...
user avatar
2 votes
1 answer
63 views

Is how I evaluated the following product correct $\prod_{n=1}^{20}(1+\frac{2n+1}{n^2})$?

The product is $$\prod_{n=1}^{20}\left(1+\frac{2n+1}{n^2}\right)$$. I can rewrite this product as $$\prod_{n=1}^{20}\left(\frac{n^2+2n+1}{n^2}\right)$$ which can be further simplified to $$\prod_{n=1}^...
user avatar
  • 2,620
4 votes
2 answers
152 views

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 ...
user avatar
  • 2,620
0 votes
0 answers
45 views

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 ...
user avatar
  • 31
2 votes
2 answers
135 views

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?
user avatar
  • 923
1 vote
3 answers
132 views

How to compute $1 \times 2 \times 3 \times 4 + 3 \times 4 \times 5 \times 6 + ... + 97 \times 98 \times 99 \times 100$

How to compute $S = 1 \times 2 \times 3 \times 4 + 3 \times 4 \times 5 \times 6 + ... + 97 \times 98 \times 99 \times 100$ Was thinking $\frac{S}{24} = {4\choose 4} + {6\choose 4} + {8\choose 4} + ... ...
user avatar
  • 1,337
1 vote
2 answers
61 views

I have to show that the sum of this double series is $\frac{1}{2}$ [duplicate]

i have to solve this double series. i tried it, but i am not sure, that it is enough. $$\sum_{i=1}^{\infty} \sum_{k=1}^{\infty} \left(\left(\frac{1}{k+1} \cdot \left(\frac{k}{k+1}\right)^{i}\right) - \...
user avatar
  • 13
1 vote
1 answer
104 views

Telescoping series: $\sum\limits_{n=1}^{∞}[\tan^{-1}(2n+1)-\tan^{-1}(2n-1)]$

In this question that was asked today the OP wrote that $$\begin{align}\sum\limits_{n=1}^{∞}[\arctan(2n+1)-\arctan(2n-1)]&=\arctan\infty-\arctan 1\\ &=\frac{\pi}{2}-\frac{\pi}{4}\\&=\frac{\...
user avatar
9 votes
1 answer
248 views

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,...
user avatar
  • 4,032
1 vote
1 answer
83 views

Why is $\sum_{k=2}^{\infty} \frac{k}{k^{2}-1}=\sum_{k=2}^{\infty}\left(\frac{1}{k-1}+\frac{1}{k+1}\right)$?

I am working through the problems on this page. I am stuck on the fourth problem. In the hints they claim that $\displaystyle \sum_{k=2}^{\infty} \frac{k}{k^{2}-1}=\sum_{k=2}^{\infty}\left(\frac{1}{k-...
user avatar
  • 674
1 vote
1 answer
80 views

Evaluate:$\sum_{n=2}^{\infty}\frac{\tan \theta_{n}}{3^n\left(3-\tan^2\theta_{n}\right)}$

Evaluate:$$\sum_{n=2}^{\infty}\frac{\tan \theta_{n}}{3^n\left(3-\tan^2\theta_{n}\right)}$$ where $$\theta_{n}=\frac{\theta}{3^n}$$ and $0<\theta<\pi$ I did try to find relation between $\tan 3x$ ...
user avatar
  • 7,126
1 vote
0 answers
159 views

Alternating, Telescoping series convergence

I know the definitions and tests for convergence, my question deals with an alternating series which converge on a range. For example, we can consider the sum of the alternating harmonic series: $$ \...
user avatar
  • 666
7 votes
1 answer
186 views

How to find sum of $\frac{1}{2}+\frac{3}{8}+\frac{15}{48}+\frac{105}{384}+\cdots$?

Sum $S_n$ of this series is given by, $S_n=\frac{1}{2}+\frac{3}{8}+\frac{15}{48}+\frac{105}{384}+...$ Express $r^{th}$ term ($U_r$) in this series and hence show that,$$\frac{U_r}{U_{r-1}}=\frac{2r-1}...
user avatar
  • 1,244
6 votes
1 answer
124 views

Non-linear recurrence relation $\frac{a_n - a_{n+1}}{1 + a_na_{n+1}} = \frac{1}{2n^2}$

Problem: I need to find all sequences $\{a_n\}_{n=1}^\infty$ that satisfy the following non-linear recurrence relation: $$\frac{a_n - a_{n+1}}{1 + a_na_{n+1}} = \frac{1}{2n^2}$$ What would the general ...
user avatar
  • 871
-2 votes
2 answers
137 views

Sum the series : $\frac{1}{9\sqrt11 + 11\sqrt9} +\frac{1}{11\sqrt13 + 13\sqrt11} +\ldots$ [closed]

$$\frac{1}{9\sqrt11 + 11\sqrt9} + \frac{1}{11\sqrt13 + 13\sqrt11} + \frac{1}{13\sqrt15 + 15\sqrt13} + \ldots + \frac{1}{n\sqrt{n+2} + (n+2)\sqrt{n}} = \frac{1}{9}$$ Find the value of $n$. I got the ...
user avatar
  • 177
1 vote
2 answers
197 views

Why isn't $\sum_{n=1}^{\infty}(-1)^n$ a telescoping series?

I read that the reason $\sum_{n=0}^\infty(-1)^n$ does not converge is because the terms alternate between -1 and 0. So I realize that this sentence does answer my question, but what I'm having ...
user avatar
  • 87
1 vote
2 answers
51 views

Find the value of $(1*1)+(1*2*2)+(1*2*3*3)+$...$+(1*2$...$(n-1)*n*n)$.

Find the value of $(1*1)+(1*2*2)+(1*2*3*3)+$...$+(1*2$...$(n-1)*n*n)$ . What I Tried: I have absolutely no idea for this. I can write this as :- $$(1!*1) + (2!*2) + ... + (n!*n)$$ However, this can ...
user avatar
  • 4,061
1 vote
1 answer
73 views

Infinite seq. of reals, for every n $ \in \mathbb{N}$ :$(a_{n-1}+a_{n+1})/2\geq a_n$. Prove $ \frac{a_0+a_{n+1}}{2}\geq \frac{a_1+a_2+...+a_n}{n} $

I realized that in the case when $$\frac{a_{n-1}+a_{n+1}}{2} = a_n,$$ the seqence reduces to a sequence of natural numbers and so the inequality is trivially true: $$ \frac{a_{n+1}}{2}=\frac{a_1+a_2+.....
user avatar
1 vote
5 answers
120 views

The Finite Sum $\sum_{r=1}^{n}\frac{1}{(3r-2)(3r+2)}$ and failure to Telescope

I'm interested in when the partial fraction method of trying to get a series to telescope fails, and have alighted upon the interesting example of, $$\sum_{r=1}^{n}\frac{1}{(3r-2)(3r+2)}$$ The ...
user avatar
1 vote
1 answer
195 views

if $S=\sin x+2\sin (2x)+\cdots+n\sin nx$,$C=\cos x +2\cos (2x)+\cdots+n\cos (nx)$

if: $S=\sin x+2\sin (2x)+\cdots+n\sin nx$, $C=\cos x +2\cos (2x)+\cdots+n\cos (nx).$ prove that $4\sin^2 (x/2).S=(n+1)\sin (nx)-n\sin(nx+x)$ I can solve this easily using complex numbers(ie taking $C+...
user avatar
0 votes
1 answer
71 views

If $f(r)=1+\frac 12 +\frac 13+..+\frac 1r$ and $\sum_{r=1}^n (2r+1)f(r)=P(n)f(n+1)-Q(n)$, where $P$ and $Q$ are polynomial functions.

If $$f(r)=1+\frac 12 +\frac 13+..+\frac 1r$$ and $$\sum_{r=1}^n (2r+1)f(r)=P(n)f(n+1)-Q(n)$$ where $P$ and $Q$ are polynomial functions, Prove that $$\sum_{r=0} ^{10} P(r)=506$$ $$\sum_{r=0}^{\infty}...
user avatar
  • 6,010

1
2 3 4 5 6