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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Summation with inner products: properties and rearrangement
OPTION 1.
I have this expression,
$$\sum \limits_{l=1}^{L}a_l^2 + \sum \limits_{l<j}^{L}a_l^2 \frac{a_j}{a_l}$$
and I would like to take out $$\sum \limits_{l=1}^{L}a_l^2$$, as to obtain something ...
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Generalisation of a multiple summation involving $i<j<k$
I've been looking at a certain type of sum, the first case being $\sum_{i<j} {a_i a_j}$, and trying to simplify it/generalise it. It would be far more useful to write them more explicitly, in terms ...
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Summation of Roots of Cubic Equation
I was attempting the recent May/June 2023, CAIE past paper for Further Mathematics variant 12. The question states:
2 The cubic equation $x^3+4x^2+6x+1=0$ has roots $\alpha$, $\beta$, $\gamma$. ...
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Evaluation of an indexed sum. [closed]
I am going through a textbook solution to an induction problem and I'm not sure why it says this summation holds.
$\sum_{m=0}^0 \sum_{k=0}^m a_{m - k, k} = a_{0, 0}$
This is very simple I know, but ...
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Efficient calculation for Lerch Transcendent Expression
I've encountered:
$$\Phi(z, s, \alpha) = \sum_{k=0}^\infty \frac { z^k} {(k+\alpha)^s}.$$
When trying to compute:
$$\frac{1}{x}\sum_{p=0}^m \frac{2}{(2p-1)\ x^{2p-1}}\ s.t. x\in\mathbb{N} =\ ???$$ ...
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Is there an identity for $\sum_{k=0}^{n-1}\csc(w+ k \frac{\pi}{n})\csc(x+ k \frac{\pi}{n})\csc(y+ k \frac{\pi}{n})\csc(z+ k \frac{\pi}{n})$?
What I'd like to find is an identity for $$\sum_{k=0}^{n-1}\csc\left(w+ k \frac{\pi}{n}\right)\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)\csc\left(z+ k \frac{\pi}{n}\right)$$...
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Follow-up question on Abel Summation
This is a follow-up to a much simpler question I asked here, which @PrincessEev answered promptly and perfectly. She showed me how to rewrite the sum $\sum _{i=1}^x \phi (x-i)$ in such a way that Abel ...
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Can I write an Abel Summation Formula for this?
Given appropriate constraints, and a continuously differentiable real-valued function $\phi (x)$, the Abel Summation Formula (Wikipedia article here) can be written as
$$\sum _{k=1}^x \phi (k) = \...
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Fubini theorem in non classical summation methods
I am aware that in the theory of classical infinite sums , one can not generally interchange the order of a double sum or do other infinite sum manipulations. However, these infinite sum manipulations ...
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How do I convert a function to sigma notation when there is a nested and lagging summation within the function?
I am trying to simplify the following function into summation form: $$f(x) = \frac{x_1}{x_0}+\frac{x_2}{x_0-x_1}+\frac{x_3}{x_0-x_1-x_2}+...+\frac{x_n}{x_0-x_1-x_2-...-x_{n-1}}.$$
However, I am unsure ...
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Is there an identity for $\sum_{k=0}^{n-1}\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)$?
What I'd like to find is an identity for $$\sum_{k=0}^{n-1}\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)$$
here it can be shown that where $x=y$,
$$n^2 \csc^2(nx) = \sum_{k=0}^...
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Symmetric sum of $ab(3a+c)$
I'm having difficulty with determining the $\sum_{sym}ab(3a+c)$ involving $3$ variables $a,b,c$. I know the cyclic sum is going to be $ab(3a+c)+bc(3b+a)+ac(3c+b)$, but I'm confused about the symmetric ...
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If $\sum_{p=0}^{2020}{\sin(2^p\theta)\sec(3^{p+1}\theta})=a\tan(b\theta)+c\tan(d\theta)$, find $ac,ad,cd,bd$
If $\sum_{p=0}^{2020}{\sin(2^p\theta)\sec(3^{p+1}\theta})=a\tan(b\theta)+c\tan(d\theta)$, find $ac,ad,cd,bd$
I converted secant to cosine, opened the sigma and tried to take LCM pairwise. But couldn'...
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How express 1234…n as a sum
I found $12345\ldots n$ is equal to $\sum_{k=1}^{n-1}k\times10^{n-k}$ when $n\le9$ $(n\in\mathbb{N})$
example) $ n=5\rightarrow1234\ldots 5 $ is 12345 = $\sum_{k=1}^{5-1}k\times10^{5-k}=1\times
...
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How can I write the following series in summation notation?
I am writing up a report for my linear algebra class where I am using series to describe certain transformations in a particular vector space:
One example is:
$$t_2 + t_5 + t_8 + ... + t_{n-1} = \sum_{...
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Summation with products of values with different powers
For an exercise I am working on I have the following equation,
$$
c\sum_{n=1}^{\infty}n\sum_{j=1}^{n}(\frac{\lambda}{2\mu})^{j}(\frac{\lambda}{3\mu})^{n-j},
$$
where $\lambda<\mu$. I have been ...
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Sum of a sum: $\sum_{n=1}^{11}n\left[\frac{1^2}{1+n}+\frac{2^2}{2+n}+\cdots+\frac{11^2}{11+n}\right]$
The given sum is:
$$\sum_{n=1}^{11}n\left[\frac{1^2}{1+n}+\frac{2^2}{2+n}+\cdots+\frac{11^2}{11+n}\right]$$
I tried simplyfing it as:
$$1^2\left[\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{12}\right] + 2^...
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What computational shortcut finds the sum all possible products given any list of n random real numbers taken r at a time? Here's what I tried...
I have the following computational shortcuts for any list of $n=4$ quantities taken r at a time. My goal is to do this for lists of any length. Taken r at a time, what function can similarly output ...
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Analytical evaluation of infinite series
I am trying to calculate the infinite series
$$\sum _{n=-\infty }^{\infty } \frac{(-1)^{n+1} e^{-(n-1)^2\pi}}{1-e^{ (2 n-1)\pi}}\simeq -0.0903244354808$$
Are there any any analytical methods to ...
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How to express this into a double summation format
Suppose, I am having difficulties writing down as double summation. The expression is $$b_1(t_n)\sum\limits_{k = 1}^1 {a_{1k}u_k}+b_2(t_n)\sum\limits_{k = 1}^2 {a_{2k}u_k}+ ... +b_p(t_n)\sum\limits_{k ...
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Summation of $\sum_{r=1}^n \sin(2r-1)\theta$
In a CIE A Level Further Mathematics question paper, the following question appeared:
By considering $$\sum_{r=1}^n z^{2r-1}$$ where $z=\cos\theta +i\sin\theta$, show that if $\sin ≠ 0$,
$$\sum_{r=1}^...
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How to prove identity $\sum_{k=0}^{n-1}\tan^2\left(\theta+{k\pi\over n}\right)=n^2\cot^2\left({n\pi\over2}+n\theta\right)+n(n-1)$?
Looking at Jolley, Summation of Series, formula 445:
$\sum_{k=0}^{n-1}\tan^2\left(\theta+{k\pi\over n}\right)=n^2\cot^2\left({n\pi\over2}+n\theta\right)+n(n-1)$
How can one prove this?
Considering $\...
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Summation in closed form
I have the following summation:
\begin{equation}
f(k_1,k_2) = \sum_{l_1 = 0}^{k_1}\sum_{l_2 = 0}^{k_2} {\rho}^{l_1+l_2}\delta[k_1-k_2+l_2-l_1]
\end{equation}
where $\rho$ is a constant and $\delta$ is ...
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Is there an identity for $\sum_{k=0}^{n-1}\tan^4\left({k\pi\over n}\right)$?
Is there a simple relation for $\sum_{k=0}^{n-1}\tan^4\left({k\pi\over n}\right)$
like there is for $\sum_{k=0}^{n-1}\tan^2\left({k\pi\over n}\right)$?
Looking at Jolley, Summation of Series, formula ...
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limit of $\sum_{k=0}^{n} q^{(2k+1)^{3/2}}$ with $0<q<1$?
It is easy to prove that for $0<q<1$, $\sum_{k=0}^{n} q^{(2k+1)^{3/2}}$ converges because it is always increasing and it is always smaller than the convergent series $\sum_{k=0}^{n} q^{2k+1}$ ...
user1096364
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How to calculate this double series?
I am doing a problem that needs me to solve the Laplace's equation in a cubric boundary condition using separation of variables. During the procedure solving it, I meet the following series:
$$\...
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Is there a formula for $\sum_{i=1}^n a^{1/i}$?
Just what the question says. $\sum_{i=1}^n a^{i}$ has quite well known equation (geometric series). I couldn't find any resources for power $ {1/n} $ though.
On a slight tangent; I tried solving it ...
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How can I intuitively get from the below summation to a generating function without knowing key identities before hand?
In class, our professor was very adamant that the following simplification is intuitive:
\begin{align*}
\sum^{\infty}_{n=0}x^n\binom{2n}{n}=\frac{1}{\sqrt{1-4x}}
\end{align*}
I can get from the RHS ...
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Showing $\lim_{x\rightarrow 1^-} \sum_{k=0}^\infty (-x)^k (2k+1) \ln(2k+1) = \frac{-2C}{\pi}$
While I was playing around with divergent summation, I noticed that the following appears to be true:
$$\lim_{x\rightarrow 1^-} \sum_{k=0}^\infty (-x)^k (2k+1) \ln(2k+1) = \frac{-2C}{\pi}$$
where $C = ...
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Borel summation of a divergent series as the best estimate of a function
My current research has me working with series of the form $\sum_{n=0}^\infty f_n z^n$, $z\in\mathbb{C}$ that usually have zero convergence radius (they diverge for all values of $z$). Due to ...
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Summation Notation for Current Index
I have a question on how to properly present a summation with the use of variables. I am not so familiar with how to properly use summation notation, and am looking for some advice on how to do so ...
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Watson-Nevanlinna theorem for $e^{-1/z}$
I am currently trying to understand Watson-Nevanlinna (WN) theorem, which gives sufficent conditions for a function $f(z)$ to be equal to the Borel sum of its asymptotic expansion as $z\to0$.
The ...
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Finding the sum of finite geometric series
I'm doing the following summation $\sum_{l=k}^{n}2^l$
$\sum_{l=k}^{n}2^l = 2^k + 2^{k+1} + 2^{k+2} + \ldots+ 2^{n-1} + 2^{n}$
$S_n=a_1\dfrac{1-r^n}{1-r} \therefore S_n=2^k\dfrac{1-(2)^n}{1-2} = 2^{k+...
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How to factorize and solve equations with $\Sigma$ notation?
I have a few doubts about the properties of sigma notation, $\Sigma$ . My questions rely on factorization and solving equations with $\Sigma$.On account of the fact that my questions are correlated, I ...
user1056027
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How do I solve the double summation $ \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{m^2 - n^2}{(m^2 + n^2)^2}$?
Basically I'm stuck with this double summation. I want some help evaluating this summation.
$$ \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{m^2 - n^2}{(m^2 + n^2)^2} $$
Am I allowed to change the ...
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How does the divergent sum $\sum_{n=1}^\infty\cos(2n\gamma)\sin(2nt)$ correctly evaluate an integral? Surely distributions don’t apply here
$\newcommand{\d}{\,\mathrm{d}}\newcommand{\res}{\operatorname{Res}}$Note: I don’t know any distribution theory myself, but I was informed by someone else and hinted to by this answer that my problem ...
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Closed form of $\sum _{k=1}^{n }\:\lfloor{n/k}\rfloor$ [duplicate]
I came across a question in which, if I am able to calculate this sum,
$\sum_{k=1}^{n }\:\lfloor{n/k}\rfloor=$ ?
it would get solved quite easily. I have never seen any closed form for this question,...
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Proving $\sum_{k=1}^{n}\frac{1-a\cos\left(\frac{2\pi k}{n}\right)}{1-2a\cos\left(\frac{2\pi k}{n}\right)+a^{2}}=\frac{n}{1-a^{n}}$
How do prove this relation?
$$\sum_{k=1}^{n}\frac{1-a\cos\left(\frac{2\pi k}{n}\right)}{1-2a\cos\left(\frac{2\pi k}{n}\right)+a^{2}}=\frac{n}{1-a^{n}} \text{ } \text{ } \text{ } \text{ } \text{ }\...
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Notions of convergence of real sequences. Where can I learn about this?
I am looking some good references to introduce myself to convergence process of real sequences. I know classical convergence, uniform convergence, almost convergence, matrix summation method, ...
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Is it possible to rewrite this sum in terms of some power series?
Is it possible to rewrite this sum in terms of some power series, maybe some cosine power series?
$$\sum_{n=0}^{\infty} \dfrac{x^{2n}}{2^{2n}(n!)^2}$$
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When calculating infinite sums, if and when applicable can you cancel out the variables to equal zero?
When calculating infinite sums, if and when applicable can you cancel out the variables to equal zero?
For example:
if I was trying to solve the infinite sum of this equation:
$$\{x + y + (-x) + (-y)\...
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Steps to Summation closed form [duplicate]
I have $$\sum_{n=1}^k 2^n$$
I got this result from trial and error (validated online), but I want to understand the steps to get there. $$= 2^{k+1}-2$$
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Is there a way to expand a summation of product of two functions?
I have been experimenting with summations were you multiply to functions, e.g.
$\sum_{i=0}^n f(n)g(n)$ where $f(x)=x^2$ and $g(x)=-\frac{x^2}{\log{x}}$. While trying out different functions for $f(x)$ ...
user1025147
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Analogue of Borel transform for holomorphic functions
Let $f$ be holomorphic at $z_0$, and therefore agrees with its Taylor series in a neighbourhood around $z_0$,
$$f(z)=\sum_{n=0}^\infty \frac{f^{(n)}(z_0)}{n!}(z-z_0)^n. $$
Is there an (integral) ...
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Proof: Sum of minimum of two functions vs minimum of sum of functions
I wonder whether the following is true:
$$\min\big\{f_1(x),g_1(x)\big\} + \min\big\{f_2(x),g_2(x)\big\}\le \min\big\{f_1(x)+f_2(x),~g_1(x) + g_2(x)\big\}.$$
I already know how to prove that:
$$\min\{f(...
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Finding the sum of geometric progression
Evaluate the sum:
$$\sum_{x=0}^\infty x(x-1) {2+x \choose x}(0.008)(0.8)^x $$
I was able to make this into:
$$0.004\sum_{x=0}^\infty x(x-1) (x+1)(x+2)(0.8)^x $$
Let $x=n-2$ then $n=x+2$:
$$0.004\sum_{...
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Square of sum of sum of variables [closed]
I've been trying to find the expansion to the expression given below for some time now.
$$\left(\sum_{i=1}^{N-1}\sum_{j=i+1}^Na[i]a[j]\right)^2$$
So what I am looking for is something like this:
$$\...
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Trying to prove $\left(1-\frac13+\frac15-\frac17+\cdots\right)^2=\frac38\left(\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\cdots\right)$
It has been more than 7 days I have been trying to prove this following result using Harmonic Numbers
Let me add this
Proving $\left(1-\frac13+\frac15-\frac17+\cdots\right)^2=\frac38\left(\frac1{1^2}+\...
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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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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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