Questions tagged [summation-method]

Use for methods for constructing generalized sums of series, generalized limits of sequences, and values of improper integrals.

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51 views

Change summation order

I need to change the summation order in the sum $$ \sum_{m,n=0}^\infty \left( \sum_{l=0}^{n+m} \left( \sum_{t=0}^l \binom{m}{t} \binom{n}{l-t} a_{n-l+2t,m+l-2t} \right) \right) \frac{ b_{m,n}}{m! n!...
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15 views

Formula for Riemann sums when subintervals aren't equal???

So I have a question about the formula for Riemann sums when the subintervals aren't equal. My textbook spent a whole two pages introducing the standard notion of a Riemann sum (when the subintervals ...
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1answer
81 views

Closed form of $\sum_{k=1}^{n} \frac{{n}\choose{k}}{k}$.

I would like to ask if it is possible to find a closed form of the sum $\sum_{k=1}^{n} \frac{{n}\choose{k}}{k}$ (1). I managed to show that it is enough to find a closed form of $\sum_{k=1}^{n} \frac{...
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3answers
69 views

How to change order of sum?

I need change the summation order in the double sum $$ S_{m,n}=\sum_{j=0}^m \sum_{k=0}^n a_{j,k} x^{j-k} B_{m+n-j-k}, $$ to separate $B$ and get somethink like to $$ S_{m,n}=\sum_{s=0}^{m+n} \left( ...
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0answers
52 views

Any ideas on how to evaluate this sum? [closed]

Have you got any ideas on how to evaluate this? $$(x+0)^{20}+(1+x)^{20}+(2+x)^{20}...+(100+x)^{20}$$
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1answer
61 views

A summation similar to Vandermonde identity

I met an identity, similar to Vandermonde's identity, but not sure how to prove: $$\sum_{j=0}^k{k \choose j}{\frac{1}{2}j \choose n}(-1)^{n+k-j}=\frac{k}{n}(-1)^k2^{k-2n}{2n-k-1 \choose n-1}, \ n \geq ...
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1answer
39 views

how can I write this sum in sigma notation? [closed]

I find it difficult to write this in sigma notation. I tried but couldn't figure out. $$ \frac{1}{n} \sqrt{1-\left(\frac{0}{n}\right)^2} + \frac{1}{n} \sqrt{1-\left(\frac{1}{n}\right)^2} + \dots + \...
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2answers
134 views

A summation identity

I have encountered this identity in Page 616 of Mathematical Methods for Students of Physics and Related Fields (Second Edition) by Sadri Hassani: $\sum_{m = 0}^{n} (-1)^m \frac{(2n + 2m)!}{(n + m)! (...
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28 views

What kind of terms dominate this summation? : Specific question

Here is my summation term. $\Sigma_{l=0}^{np}\langle e_{1}|f_{1}\rangle^{l}\langle e_{1}|f_{2}\rangle^{np-l} \langle e_{2}|f_{1}\rangle^{nq-l} \langle e_{2}|f_{1}\rangle^{n(1-q)-np+l} {nq\choose l} {n(...
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41 views

Is the Cesàro sum of the Fourier series of $f$ a best approximation of $f$ in any sense?

Let $(e_n)_{n\in\mathbb{\mathbb{Z}}} = (t\mapsto e^{int})_{n \in \mathbb{Z}}$ the Fourier orthonormal base of $L^2(\mathbb{T})$ with the scalar product $\langle f,g\rangle = \int_{-\pi}^\pi f(t)\bar{g}...
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2answers
23 views

Using the sum disturbance method, find a compact form of the following sums:

I tried to solve these two examples, but without success, could someone help me solve it because I got stuck on them and don't understand how to solve them. Using the sum disturbance method, find a ...
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1answer
44 views

How do I prove that sum of reciprocals of first $2^n$ natural nos is always greater than $\frac{n+1}{2}$

How can prove this inequality \begin{equation} \sum_{r=1}^{2^n} \frac{1}{r}\geq \frac{n+1}{2} \end{equation} Without using induction... Want to have an insight on summation inequalities
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1answer
55 views

Evaluating $\sum_{r=1}^{2001} f\left(\frac{r}{2002}\right)$, where $f(x)=\frac{4^{x}}{4^{x}+2}$ [duplicate]

Let $ f(x)=\dfrac{4^{x}}{4^{x}+2} $. Find $ \sum_{r=1}^{2001} f\left(\frac{r}{2002}\right) $. Given, $$f(x)=\frac{4^{x}}{4^{x}+2}$$ $$ \begin{align} \sum_{r=1}^{2001} f\left(\frac{r}{2002}\...
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35 views

How to find specific solutions?

Show that $\left[\frac{\sum_{r>s} x^{r}}{r !}\right] \div\left[\frac{\sum_{r>s} y^{r}}{r !}\right]>\frac{x^{s}}{y^{s}},$ whenever $x>y>0$ MY APPROACH: $$ \begin{array}{l}r=5, \quad s=4 ...
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1answer
44 views

Theoretical Procedure for Power Series Equation:

If I have the following equation: \begin{equation}2c_0(x-1)+\sum_{k=2}^\infty[(c_{k-2}+2c_{k-1})(x-1)^k]+\sum_{k=0}^\infty[(c_{k+2}(k+2)(k+1)+kc_k+(k+1)c_{k+1}+c_k)(x-1)^k]=0 \end{equation} I was ...
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1answer
14 views

Given the following summation is there a way to combine given the following orientation?

I have the following summation: \begin{equation}\sum_{n=2}^\infty c_n(n)(n-1)(x-1)^{n-2}+(x+1)\sum_{n=1}^\infty nc_n(x-1)^{n-1}+\sum_{n=1}^\infty nc_n(x-1)^{n-1}+(x+1)^2\sum_{n=0}^\infty c_n(x-1)^n +2(...
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1answer
110 views

Approximation with error term / bounds for double summation

I'm looking for ways to find continuous functions that approximate double summations of the form $S(n)=\sum _{j=1}^n \sum _{i=1}^n f(n-i j)$ for functions $f:\mathbb{R}\to \mathbb{R}$, and for large $...
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3answers
54 views

Summation problem: find k if $\sum_{k=5}^{29} kn-6=1125$ [closed]

How do I find k if $\sum_{k=5}^{29} kn-6=1125$ ? I tried to solve it but couldn’t understand it. Any hints would be appreciated!
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1answer
57 views

General formula for the sum of x raised to general degree: $(1^z + 2^z + \cdots+ x^z)$

As I was reading a book on the financial market micro-structure, I came across a simplification that I have not been able to prove. The book states that $\sum_{\ell=1}^{Q}2G_0(\frac{1+\gamma}{\ell})\...
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1answer
20 views

Theorem needed to prove a summation

I know that the following relation holds: $$\sum_{x=1}^y\frac{x(5x+6)}{45}=\frac{y(y+1)(10y+23)}{270}$$ But what theorem should I use to prove that relation?
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48 views

Closed form for $\sum_{n=1}^{\infty} \sum_{k=1}^{n-1} \frac{(-1)^n}{k(n-k)}$

Mathematica gives $$\sum_{n=1}^{\infty} \sum_{k=1}^{n-1} \frac{(-1)^n}{k(n-k)}=0.480453...$$ The question is: Can one get a closed form for this summation by hand?
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37 views

Find the numbers that would satisfy an upper index, that would lead it to be divisible by 3

\begin{equation}\displaystyle{\sum_\limits{i=0}^n}i\end{equation} Find the values of n for which the summation is divisible by 3. If Possible write a general formula.
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46 views

Fast evaluation of sums of products of two sinusoidal functions

$s_1, s_2,\ldots,s_N$ is a set of real numbers. It is required to evaluate the sums \begin{eqnarray} S_1(n_1,n_2)&=&\sum_{j=1}^N \cos(n_1 s_j)\cos(n_2 s_j)\nonumber\\ S_2(n_1,n_2)&=&\...
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3answers
82 views

Finding the Sum of series $S = \sum_{n=1}^{243} \frac{1}{n^{4/5}} $.

If $S = \sum_{n=1}^{243} \frac{1}{n^{4/5}} $. Find the value of $\lfloor S \rfloor$ where $\lfloor \cdot \rfloor$ represents the greatest integer function. By approximation using definite integral, ...
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1answer
44 views

Geometric series with product of binomial coefficents

I'm struggling with the following summation $$F(x)=\sum\limits_{t=1}^{\infty}\binom{-5/2}{2t}\binom{2t}{t-1}x^{t}$$ I have tried to look for ways to reduce the product of the binomial coefficient to ...
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1answer
50 views

sum of power floor $ S(n,m) = \sum_{i = 1}^{n} \Bigl \lfloor \frac{n}{i^m} \Bigr \rfloor i^m $

How do I calculate $$ S(n,m) = \sum_{i = 1}^{n} \Bigl \lfloor \frac{n}{i^m} \Bigr \rfloor i^m $$ This can be simplifies to $$ S(n, m) = \sum_{i = 1}^{\sqrt[m] n} \Bigl \lfloor \frac{n}{i^m} \Bigr \...
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1answer
169 views

Finding $Z(N)=\sum_{i=1}^N i^2\left\lfloor \frac{N}{i^2} \right\rfloor$ [closed]

I have been trying to find this summation faster, Is there any sequence that can be observed? $$Z(N)=\sum_{i=1}^N i^2\left\lfloor \frac{N}{i^2} \right\rfloor$$
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1answer
29 views

Summation of $\frac{n^2}{(n^2-4)^2}$ where $n$ runs over odds

I want to evaluate the following sum over odds: $$\sum_{n=1,3,5,\dots}\frac{n^2}{(n^2-4)^2}.$$ I tried to set $n=2k+1,k=0,1,2,\dots$ but that makes it complicated.
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2answers
76 views

Double Summation indexes problem

I have the following sum: \begin{equation} \sum_{j=0}^{a} \sum_{k=0}^{n-2j} c_{jk}\,\, x^{\,j+k} \end{equation} Where $a=\lfloor n/2\rfloor$. I want to convert the previous sum to other like: \...
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0answers
21 views

Proof for Limitation Theorem for Euler-Summation

I'm currently stuck in Hardy's Book 'Divergent Series' at a very small Proof which i can't seem to grasp. Here, $(E,q)$ Summation is defined over $\lim_{m \rightarrow \infty} A_m^{(q)}$ or $\sum_{n=0}^...
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74 views

Is there a series representation for $\frac{1}{\log(x)}$?

for $x>0$ it is known that: $$\log(x)=2\sum_{k=1}^\infty \frac{\frac{x-1}{x+1}^{2k-1}}{2k-1}$$ Is there a series representation for $\frac{1}{\log(x)}$ in the following form? $$\frac{1}{\log(x)}=...
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1answer
31 views

Closed form for the function $G(x) = \sum_{n=1}^\infty (e^{x/n²}-1) $

Consider the following function : $$G(x) = \sum_{n=1}^\infty (e^{x/n²}-1)$$ I know that the sum converges . Also ,$$G(x) =(1/2π) \int_{-π}^{π} f(xe^{-it})e^{e^{it}} dt$$ Where, $f(x)= π\sqrt{x}...
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1answer
31 views

How do I evaluate this summation series?

I came across this problem and I couldn't solve it $\sum_{r=1}^\infty\frac{6^r}{(3^r-2^r)(3^{r+1}-2^{r+1})}$ So when I saw the solution they wrote $6^r = 3^r(3^{r+1}-2^{r+1}) - 3^{r+1}(3^r-2^r)$ and ...
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1answer
21 views

What does this double summation with mod evaluate to?

Let $X = \{1,2,3,\dots\}$ and $Y = \{0,1\}$. Define $f:X\times Y \rightarrow \mathbb{R}$ by $$ f(x,y) = \begin{cases} -2^{-x} &\text{ if } \mod(x, 2) = y \\ (y+1) 2^{-x} &\text{ if } \mod(x,...
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1answer
25 views

This Summation yields different results when a specific identity from Combinatorics is used.

At first I had this summation : $$ \sum_{m=1}^{n} (m)\binom{n}{m}(3^m -2^m) $$ After that I used the identity $\binom{n}{r}=\frac{n}{r}\binom{n-1}{r-1}$ Which gave me the modified form of the ...
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1answer
27 views

Decreasing summation amount based on number selected

I am working on creating a calculated field on a form which has limited mathematical capabilities. My available operators are: + - / * ( ). My calculated field is ...
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2k views

Towards a new proof of infinitude of primes ( with possible unified application to other primes of special forms whose Infinitude is unknown):

I'm trying to prove the infinitude of primes as follows: Consider the following partial sum : $$S(p)=\sum_{n=2}^p\sin^2\left(\frac{π\Gamma(n)}{2n}\right)$$ The summand is zero for non-primes greater ...
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19 views

Applications of divergent series

There are certain evaluation methods like Cesaro, Abel, and Ramanujan summation, and other techniques like analytic continuation of the zeta function that can be used to assign values to divergent ...
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23 views

Closed form expression for the following Binomial expansion

I have the following Binomial expansion which I want to bound tightly. I have been able to bound but it is very loose. My method is as follows :- $2\sum_{i=1}^{k/2} {n \choose n/2 +i} + {n \choose n/...
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33 views

$\sum_{(s,t)\in S\times T} f(s,t)=\sum_{s\in S}\sum_{t\in T} f(s,t)$ via Fubini-Tonelli?

Let $f: S\times T \to [0,\infty[$ be a function. I wish to show that $$\sum_{(s,t)\in S\times T} f(s,t)=\sum_{s\in S}\sum_{t\in T} f(s,t)$$ I know that in general we have $$\int_X g d\mu =\sum_{x\...
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3answers
114 views

A closed form for $\sum_{k=0}^n \frac{ (-1)^k {n \choose k}^2}{k+1}$

Mathematica gives $$\sum_{k=0}^n \frac{(-1)^k {n \choose k}^2}{k+1}= ~_2F_1[-n,-n;2;-1],$$ where $~_2F_1$ that is Gauss hypergeometric function. Here the question is: Can one find a simpler closed ...
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1answer
114 views

Show $\sum_{n \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} a_k b_{n-k} z^n =(\sum_{n \in \mathbb{Z}}a_n z^n)(\sum_{n \in \mathbb{Z}}b_n z^n)$

First, let me tell the definition of series I use. Let $S$ be any set. Let $f: S \to \mathbb{C}$ be a function. We say $\sum_{n \in S}f(n)$ converges to $F\in \mathbb{C}$ if the following condition is ...
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0answers
16 views

Add function value over an interval.

To prove: $$\sum_{i=1}^{S}I_i=S \int_{\underline{I}}^{\overline{I}}I f(I)dI$$, where $f(I)$ is a continuous probability density function and $I\in[\underline{I},\overline{I}]$ and $S$ is population . ...
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2answers
66 views

Estimate on partial sum $\sum_{n=1}^x \frac{\sin^2(n)}{n}$ using Abel Plana Summation Formula :

I'm trying to get estimate on the following partial summation using Abel-Plana Summation formula: $$\sum_{n=1}^x \frac{\sin^2(n)}{n}$$ I can handle the first integral in the formula but I'm stuck at ...
19
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4answers
697 views

On proving that $\sum\limits_{n=1}^\infty \frac{n^{13}}{e^{2\pi n}-1}=\frac 1{24}$

Ramanujan found the following formula: $$\large \sum_{n=1}^\infty \frac{n^{13}}{e^{2\pi n}-1}=\frac 1{24}$$ I let $e^{2\pi n}-1=\left(e^{\pi n}+1\right)\left(e^{\pi n}-1\right)$ to try partial ...
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1answer
16 views

How is $\sum_{k \geq 0} \sum_{i \geq k+1} \mathbb{P}(X=i) = \sum_{i \geq 1} \sum_{k=0}^{i-1}\mathbb{P}(X=i) = \sum_{i \geq1} i \mathbb{P}(X=i) $?

EDIT: Here's a way to remember without thinking hard about it that I came with from the channel of blackpenredpen: $\sum_{k \geq 0} \sum_{i \geq k+1} \mathbb{P}(X=i) $ We have $1\leq \underline{k+1 ...
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1answer
116 views

$ \sum_{n = 1}^{\infty} ( \frac{p_{2n-1}}{p_{2n}} - \frac{p_{2n}}{p_{2n +1}} ) = ?? $

Let $ p_n $ be the $n$ th prime. I was confused about the following idea. $$A = \sum_{n = 1}^{\infty} ( \frac{p_{2n-1}}{p_{2n}} - \frac{p_{2n}}{p_{2n +1}} ) $$ Very confused actually. Does this ...
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1answer
62 views

Prove the combinatorial identity

For any $n$ prove the following identity $$ \sum_{i+j+k=n} j \binom{n}{i,j,k} a_{2i+1,2j-1,2k}=\sum_{i+j+k=n} i \binom{n}{i,j,k} a_{2i-1,2j+1,2k}, $$ here $$\binom{n}{i,j,k}=\frac{n!}{i! j! k!}, $$ is ...
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1answer
64 views

question deleted as question is incomplete. [closed]

question deleted as question is incomplete.
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1answer
23 views

what will be the $(t+1)$-th term in this summation expansion and why? [closed]

What will be the last term in the following summation $$\sum_{k=0}^{t}F_{t}F_{t-1}\cdots F_{k+1}x_k.$$

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