Questions tagged [telescopic-series]
By Daniel Johnston
Ask Question
For summation questions involving telescopic sums/series. This tag is often used with (summation) or (sequences-and-series).
270 questions- Bountied 0
- Unanswered
- Frequent
- Score
- Unanswered (my tags)
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 $$\sum_{k=0}^n \binom{n}{k} = 2^n$$ ... summation algorithms binomial-coefficients hypergeometric-function telescopic-series- 2,447
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 ... algebra-precalculus contest-math recurrence-relations telescopic-series- 486
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: $\sum_{t=1}^T \Big( ||x_t - x^*|... inequality machine-learning gradient-descent telescopic-series- 609
Summation of alternating series:$\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 ... sequences-and-series summation telescopic-series- 38k
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 ... sequences-and-series summation telescopic-series- 45
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} ... integration trigonometry indefinite-integrals telescopic-series- 5,342
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 ... discrete-mathematics summation telescopic-series- 87
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 ... sequences-and-series algebra-precalculus telescopic-series- 1
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 ... calculus sequences-and-series limits telescopic-series- 215
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,... sequences-and-series self-learning telescopic-series- 315
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 ... calculus sequences-and-series factorial telescopic-series- 101
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. sequences-and-series analysis geometric-series telescopic-series- 13
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 ... sequences-and-series algebra-precalculus limits telescopic-series- 1,260
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 ... real-analysis sequences-and-series limits convergence-divergence telescopic-series- 55
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}}{... sequences-and-series inequality summation telescopic-series- 339
15 30 50 per page12345…18 Next
