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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70 views

write a closed form of $\sum_{1=k}^n i/2^i$

Ive been thinking about this problem for a while now and i cannot understand if there is an actual typo in the equation or its supposed to be like this. I havent been given any value for i. Thats why ...
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0answers
41 views

Polygamma sum problem ...

Hello guys i have a problem evaluating the following sum $$\sum_{n=1}^{+\infty}\frac{n(n+1)}{2}\frac{4x(3\pi ^{2}(n+1)^{2}+x^{2})}{(x^{2}-\pi ^{2}(n+1)^{2})^{3}}$$ It is obviously of the polygamma ...
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1answer
34 views

Polygamma sum problem

I have a problem evaluating the following sum, $$\sum_{n=1}^{+\infty}\frac{4nx(3\pi ^{2}(n+1)^{2}+x^{2})}{(x^{2}-\pi ^{2}(n+1)^{2})^{3}}$$ The sum obviously is of the form of a polygamma function. ...
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Apply the Abel summation formula to $\sum_{i=1}^k \operatorname{sinc} \bigl( \pi (x-i p) \bigr)$

As part of an exploration of the Abel Summation formula (see here), I am looking at an impulse train $T$ made up of $k$ $\operatorname{sinc}$ pulses at intervals $p$ along the $x$ axis: $$T(x):=\sum_{...
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Change of variables for summation fails

Suppose I have the sum $ \displaystyle S_1 = \sum_{i=1}^5 \cfrac{1}{(i+1)^2} $ I can make a change of variables by letting $ j=i+1 $. When $ i=1 \implies j=2 $, and when $ i=5 \implies j=6 $ $ \...
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3answers
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How to solve this recurrence with the iterative substituion method

edit: the reason i didn't accept any answers yet, is because we NEED to use the substituion method for this like i attempted in my post, so we have to give the first i=1 i=2 i=3 until we can figure ...
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0answers
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Closed form of geometric series [duplicate]

$ \displaystyle \sum_{k=0}^n ar^k =\cfrac{ar^{n+1}-a}{r-1} $ Is there a similar formula for $ \displaystyle \sum_{k=0}^n ar^{k^2} $
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Summation Problem, Need help to continue. [closed]

Im working on certain problem, here is my effort: My question is how do I continue this? any properties or rules could I use there? I want to get the theta result for Brute Force String Matching ...
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54 views

Expansion of Binomial Like Product

I am trying to understand the following pattern from the expansion of $$(x - a_n)(x - a_{n - 1})\cdots(x - a_0)$$ Where $a_n\cdots a_0$ are distinct coefficients, I want to find the sum of the ...
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250 views

Integral form(s) of a general tetration/power tower integral solution: $\sum\limits_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(An+B,Cn+D)}{Γ(an+b,cn+d)}$

In many tetration/power tower integrals, one sees a general form of the following. Let this new function be notation used to show the connection between the general result and special cases using ...
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Why can i move the summation sign down?

Hi, i'm trying to teach my self machine learning by going through the book "An introduction to Statistical Learning", and got stuck on one of the exercise questions. In the attached image ...
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How to calculate coefficients for cubic least squares regression utilizing equations for summation of x- and y- data points only.

For quadratic least squares statistical regression equations, the following is utilized for the calculation of coefficients $(a,b,c)$: $a = \cfrac{∑(x^2y) * ∑(xx)-∑(xy) * ∑(xx^2)}{∑(xx) * ∑(x^2x^2)-∑(...
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1answer
49 views

Solve two term recurrence relation

I want transform this recurrence relation into the closed formula, but i am stuck in some place, please provide some hints or steps to help me. The recurrence relation are $$ C_{n} = C_{n-1} + (\frac{...
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Doubt regarding Abel Plana summation formula

I have a doubt regarding the Abel Plana summation formula. What is the condition to apply the Abel Plana summation formula.?? This question arises because sometimes while evaluating series it gives ...
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Is it possible to evaluate the following limit using integration?

Let $f(k)=2^k$ and if the value of $\lim_{n\to \infty}16^{f(n)}\overset{k=n}{\underset{k=1}{\prod}}\frac{1}{(2^{2f(k)}-2^{f(k)}+1)}$ is P then find P/7. This is how I tried to solve: $$P=\lim_{n\to \...
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1answer
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How to express the following summation in matrix form?

Given data X and Y in matrix form, consider estimating 𝜷 by minimizing the follwing: $$ \sum_{i = 1}^n (y_i -x_i^T 𝜷)^2 + \sum_{j = 1}^p (𝜷_j)^2 $$ where p is the dimension of X (number of ...
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Evaluating summation involving binomial coefficients

I need to evaluate $$\fbox{$\dfrac{\sum_{k=0}^r{{n \choose k } \cdot {{n-2k} \choose {r-k}}}}{\sum_{k=r}^{n}{{n \choose k} \cdot {2k \choose 2r} \cdot \left(\frac{3}{4}\right)^{n-k} \cdot \left(\frac{...
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2answers
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How to write this sum $\sum_{m=0}^\infty (2m+1)^{-2k-1} \sum_{r=1}^\infty (-1)^{r-1} e^{-2(2m+1)r a} $ as a sum over single index?

So I want to write the sum $$\sum_{m=0}^\infty (2m+1)^{-2k-1} \sum_{r=1}^\infty (-1)^{r-1} e^{-2(2m+1)r a} $$ where $a>0$ and $k\in \mathbb{N}$, as a sum over single index which probably uses odd ...
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1answer
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Is double summation distributive?

Is double summation distributive like single summation? For example in the proof for linearity of expectations, $E(ax+by)=aE(x)+bE(y)$, it is shown using double summation and they seemed to break it ...
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2answers
40 views

Transformations on $ \sum_{k} {r \choose k}{k \choose n} (-1)^{r-k} = {0 \choose n-r} = \delta_{nr} $

This equality holds $$ \sum_{k} {r \choose k}{k \choose n} (-1)^{r-k} = {0 \choose n-r} = \delta_{nr} , $$ integer n, integer r $\geq$ 0, and $\delta$ is the Kronecker delta. ...
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In search for a counterexample related to the Abel-Stolz theorem

Edit: cross-posted to the MathOverflow with some with some modifications in order to answer to questions posed in the comments. Now it has an accepted answer by Pietro Majer and a very interesting ...
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Properties of $\sum _{l=0}^{\infty } \left( -1 \right) ^{l}{m+l-1\choose l} \left( 2 \,{\frac {l}{m}}+1 \right) ^{1-\nu}$

The sums defined by: $$ S(m,\nu):=\sum _{l=0}^{\infty } \left( -1 \right) ^{l}{m+l-1\choose l} \left( {\frac {2 \,l}{m}}+1 \right) ^{1-\nu} $$ are divergent. However they can be Abel-regularised as $$ ...
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What summation rule am I missing here?

I was trying to prove the associativity of matrix dot products and I think I almost got it till the last step where I'm stuck, probably due to my lack of knowledge about summation rules. I have $A\in\...
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1answer
54 views

Finding $\sum_{0 \le r <s \le n}~ (r+s){n \choose r}{n \choose s}$

We know that $$\sum_{0 \le r <s \le n} A_r A_s=\frac{1}{2}\left [(\sum_{t=0}^{n} A_t)^2-\sum_{t=0}^{n} A^2_t\right]$$ Nest using $A_r={n \choose r}x^r$, we can get $$S(x)=\sum_{0 \le r <s \le n} ...
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2answers
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Limit of power series equals $1-L$ [closed]

If $(a_n)_{n\ge 0}$ is a sequence of complex numbers and $\lim_{n\to \infty} a_n=L\neq 0$ then $$\lim_{x\to 1^-} (1-x)\sum_{n\ge 0} a_n x^n=L$$ EDIT: The initial question had a typo from the place I'd ...
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1answer
86 views

How to evaluate the following sum?

$\displaystyle \frac{100}{100}\cdot 1 + \frac{100\cdot 99}{100^2}\cdot 2 + \frac{100\cdot 99\cdot 98}{100^3}\cdot 3 + \frac{100\cdot 99\cdot 98\cdot 97}{100^4}\cdot 4 + \ldots + \frac{100\cdot 99\cdot ...
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Does my odd proof for the Abel sum for $\eta(-2)$ work?

EDIT: The correct answer to the Abel sum of $\eta(-2)$ has been given by the comments under this post. The focus of the question is now whether there is any sense to my method and my "proof" ...
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0answers
72 views

Partial sum estimate for $\sum_{p\leq x} \frac{1}{p^{1/2}}$

Forgive my question if its elementary, I'm very new in number theory. We know from Merten's theorem $$\sum_{p\leq x} \frac{1}{p} = \ln \ln x + B + O\left(\frac{1}{\ln(x)}\right)$$ Here $B= \sum_p \...
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1answer
86 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'...
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3answers
436 views

$\sum_{n=1}^\infty\frac{n}{(2n-1)16^n}\binom{2n}{n}^2\left(\sum_{k=n}^\infty\frac{2^k}{k\binom{2k}{k}}\right)=1-\sqrt2+\log(1+\sqrt2).$

Prove:$$\sum_{n=1}^\infty\frac{n}{(2n-1)16^n}\binom{2n}{n}^2\left(\sum_{k=n}^\infty\frac{2^k}{k\binom{2k}{k}}\right)=1-\sqrt2+\log(1+\sqrt2).$$ I'm sorry that I don't even know how to start. I haven'...
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1answer
49 views

If $S_{n,m}=\sum_{k=1}^{n} k^m =\sum_{j=0}^{m-1} A_{n,j}(m) S_{n,j},$ what are $A_{n,j}(m)$

We know the sum of first $n$ natural numbers, their squares and cubes. sum of higher powers can be worked out using the differences: $k^m-(k-1)^{m}$. However, these formulas are not remembered well. ...
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Alternative method to showing that $\sum_{r=-\infty}^{\infty}\frac{1}{64r^4+1}=\frac{\pi}{4}\frac{1+\sinh(\pi/2)}{\cosh(\pi/2)}$

In my answer to this question I proved the fact that $$\sum_{r=-\infty}^{\infty}\frac{1}{64r^4+1}=\frac{\pi}{4}\frac{1+\mathrm{sinh}(\pi/2)}{\mathrm{cosh}(\pi/2)}$$ using quite a non-advanced method*; ...
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1answer
57 views

How many real solutions does the equation $f(x) =0$ have, where $f(x) =\sum_{i=1}^{2020} \frac {i^2}{\left( x-i\right)} $

How many real solutions does the equation $f(x) =0$ have, where $f(x) =\sum_{i=1}^{2020} \frac {i^2}{\left( x-i\right)} $ $\bf{My Try} :$ This is a equation of degree 2020. Now let $\alpha$ be a ...
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5answers
155 views

How do I evaluate $\sum_{n=1}^{\infty}\frac{n}{2n-1} - \frac{n+2}{2n+3}$? [closed]

$$\sum_{n=1}^{\infty}\frac{n}{2n-1} - \frac{n+2}{2n+3}$$ I've tried combining the sum, telescoping series and even trying to make an Nth partial sum but nothing seems to budge. I'm not sure where to ...
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1answer
35 views

A converse to a lemma of Cesàro?

The following classical result is due to E. Cesàro: Let $\left\{ a_{n}\right\} _{n\geq0}$ be a sequence of complex numbers and let $\left\{ b_{n}\right\} _{n\geq0}$ be a sequence of positive real ...
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1answer
48 views

Where does the Integral went wrong .... $ \displaystyle \int _{0}^{\pi} \frac {\ln ( 1 + \sin \alpha \cos x )}{\cos x} \ \mathrm {d}x $

This is how I solved : Let $$ \displaystyle y = \int_{0}^{\pi} \frac {\ln ( 1 + \sin \alpha \cos x )}{\cos x} \ \mathrm {d}x $$ Then I used expansion of $\displaystyle \ln (1+x) = \sum _{n=0}^{\...
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1answer
63 views

Prove that the infinite sum of the difference of even and odd values of the Riemann zeta function is 1/2

I am interested in finding closed form solutions for the positive odd integers of the Riemann zeta function, of which only 1 is known. Please forgive me if this is already proven or well-known, but in ...
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1answer
77 views

What is $\sum_{3\leq p\leq x} \pi(\sqrt{p})$?

What is the $\displaystyle \sum_{3\leq p\leq x} \pi(\sqrt{p})$? I thought about starting from $\displaystyle 2\sum_{3\leq p\leq x}\frac{\sqrt{p}}{\log p}$.
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83 views

How to go from this summation to this approximation?

https://learn.fmi.uni-sofia.bg/pluginfile.php/194197/mod_resource/content/2/Telephone_numbers.pdf I'm a high school student investigating on telephone numbers (involution numbers) depicted as T(n). If ...
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1answer
33 views

Why this iteration summation is wrong when I calculate it?

$V_n = 3^{n}$ I need to calculate its summation where $S = (V_0)^{2} + (V_1)^{2} + ... + (V_{n-1})^2 $ Obviously, you can just put a new iteration $(W_n)$ where $W_n = (V_n)^{2}$ You find it $W_n = 9^{...
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1answer
36 views

Is it possible to rewrite this equation using Einstein Summation?

Can we rewrite the below linked equation as: $$\overline{x} = \frac{x_i}{n} $$ Or is it necessary to have two variables and indices? Here's the equation
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2answers
71 views

Evaluating a sum without using a program

$$ \sum_{k=1}^{\infty} \frac{e^k}{k^k} $$ The solution is about $\approx {5.5804}$ But I don't know how to calculate this sum, I tried using the squeeze theorem but I couldn't find $2$ series that ...
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0answers
48 views

Summation of an infinite series from two ways leading up to two answers

Compute the sum (S): $\frac11.\frac12 + \frac12.\frac13 + \frac13.\frac14 + \frac14.\frac15 ... infinite terms + $ $\frac11.\frac12 + \frac12.\frac13 + \frac13.\frac14 + \frac14.\frac15 ... infinite ...
6
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1answer
138 views

Are there any good examples from other math fields or intuition supporting $\int_0^1\frac1xdx=\int_1^\infty\frac1xdx$?

This question is related to the potential possibilities of classification of divergent integrals more precisely than just "divergent to infinity" and the like. Improper divergent integrals ...
3
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1answer
103 views

Euler's summability method for series convergence

I was going through the book on Functional Analysis by Erwin Kreyzig, and I came across this as one of the exercises. As an application of functionals to summability of sequences, the following ...
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1answer
58 views

How to determine asymptotics for $\sum_{ab^2 < x} ab^2$ by summing separately over $a$ and $b$

I do not understand how to get asymptotics for the double sum $$\sum_{ab^2 < x} ab^2$$ If I sum over $a$ first, I get $$\sum_{b^2<x} b^2 \sum_{a < x/b^2} a = \frac12 \sum_{b^2<x} b^2 \frac{...
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1answer
102 views

Double summation index notation: $\Sigma_{i<j}$ versus $\Sigma_{i\neq j}$?

What is the difference between the summations using $i<j$ and $i\neq j$ in the formula below: $$\sigma^{2}(\boldsymbol{w})=\sum_{i} \tilde{w}_{i}^{2}+2 \sum_{i<j} \tilde{w}_{i} \tilde{w}_{j} \...
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3answers
98 views

it's possible to invert summation/ series limits?

If the summation just sum every term i was thinking that for instance 1+2+3+4 = 4+3+2+1 so why this $$\sum\limits_{i=1}^{n} (3i)\ = \sum\limits_{i=n}^{1} (3i) $$ is not true ? And how i can ...
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1answer
43 views

Something is wrong with my proof by induction solution

I am trying to prove a formula by induction but I cant get it right. Something is wrong somewhere and I dont know where. Any help will be greatly appreciated. We want to use induction to prove that: $$...
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3answers
77 views

Closed form for $1-2+3-4+\cdots(-1)^{n-1}n$

How do I find a closed formula for the following? $$\sum_{i=1}^n(-1)^{i-1}i$$ If $n$ is odd number, I can express it as $\frac{n+1}{2}$; if $n$ is even, then the expression if $\frac{-n}{2}$, but how ...

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