Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [summation-method]

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

In mathematical analysis, the need arises to generalize the concept of the sum of a series (limit of a sequence, value of an integral) to include the case where the series (sequence, integral) diverges in the ordinary sense. This generalization usually takes the form of a rule or operation, and is called a summation method.

348 questions
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Finding the sum of finite geometric series

I'm doing the following summation $\sum_{l=k}^{n}2^l$ $\sum_{l=k}^{n}2^l = 2^k + 2^{k+1} + 2^{k+2} + \ldots+ 2^{n-1} + 2^{n}$ $S_n=a_1\dfrac{1-r^n}{1-r} \therefore S_n=2^k\dfrac{1-(2)^n}{1-2} = 2^{k+n}-2^k$ But my final result seems to be incorrect…
Carlos
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Is there some really powerful summation method?

I know that Borel summation gives you a value if the coefficients are bounded by $n!C^n$. Is there a more powerful summation method (with nice properties comparable to Borel) that sums series with coefficients bounded by $(kn)!$ or even $(n^k)!$ ?
FusRoDah
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How to add to summations with different limits

$$\sum_{k=0}^n {n \choose k} x^{n+1-k}y^k+\sum_{k=1}^{n+1} {n \choose k-1}x^{n+1-k}y^k$$ i would like to know how to add the two summations above together including a explanation of how the limits of the two sums will change.
BOW
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Cesaro summation for matrices

I wonder if Cesaro summation for matrices is the same as summation for sequences. Cesaro summation for sequences means convergence of the arithmetic means (averages) of partial sums of sequence. For the sequence $a_1, a_2, \ldots$, Cesaro summation…
Konstantin
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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 $\displaystyle 2\sum_{3\leq p\leq x}\frac{\sqrt{p}}{\log p}$.
A2011
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How to find asymptotic of $\sum_{k=0}^{\lfloor\log(n) \rfloor} \lceil \frac{n}{2^k} \rceil$

My rough estimates are the following: $\lceil \frac{n}{2^k} \rceil \le \frac{n}{2^k} + 1$, so sum $\le \lfloor\log(n) \rfloor + n*\sum_{k=0}^{\lfloor\log(n) \rfloor} \frac{1}{2^k} \le 2n(1-1/n) + \lfloor\log(n) \rfloor = O(n)$. Am I right? And are…
Alexander Slesarev
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Closed form expression of following equation

Can someone hint me on solving the following equation to find $P^*$ \begin{equation} P^* =\sum^\infty_{n=m}\{P(m).P(R_{n\mid m} \} \end{equation} Where, \begin{equation} P(m) = e^{-\lambda t}(\lambda t)^m /…
SJa
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A binomial multiplied by a shifted poisson

I am trying to simplify this expression $$ \sum_{k\geq 0} \frac{(k+x)!}{k!} \frac{b^{s+k+x}}{(s+k+x)!} $$ to an expression with finite summation. I am able to do for x = 0 and 1, but not able to reduced for x>1. If anyone has similar experience in…
S. Bha
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Solve the equation: $p = 1+r+r^2+ \ldots+r^n$

There is an equation: $p = 1+r+r^2+ \ldots+r^n$. The right side of this equation can reduce to $(r^n-1)/(r-1)$, but I cannot find the way to find a function of $r$ and $p$ such that $n = f(p, r)$, can someone know how to get this function.
Fhope Cc
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What is the difference between these two geometric formulas?

I've seen these two different geometric sum formulas but I don't know when which is used. $S = a\left(\frac{1-(r^n)}{1-r}\right)$ and $S = a\left(\frac{1-(r^{n+1})}{1-r}\right)$
Nwqp
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how can I write this sum in sigma notation?

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