Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [summation]

Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.

The notation $\sum\limits_{i=1}^na_i$ means $a_1+\ldots +a_n$.

Use for sums of infinite series and questions of convergence; use for questions about finite sums and simplification of expressions involving sums.

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Proving a complicated summation problem

I'm trying to solve a complicated summation problem where there are two multiplication problems that both have $i$. I'm entirely lost. I would be thankful for any help. How is it possible to possible to find the sum of this expression (with a…
ik629
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Proving $\sum _{r=1}^n r^4$ by considering $\sum _{r=1} ^n \left( (r+1)^5 - r^5 \right)$

By considering $$\sum _{r=1} ^n \left( (r+1)^5 - r^5 \right)$$ Show that $$\sum _{r=1}^n r^4 = \frac 1 {30} n(n+1)(2n+1)(3n^2 + 3n-1)$$ So I'm a little unsure on how to really start on this so I've expanded the given expression an found that…
H.Linkhorn
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Reducing this sum to closed form

I am having some difficulty finding a closed-form for this $\displaystyle\sum_{x=1}^{L}(L+1-xa)(L+1-xb)(2^{x-1}-1)$ (assume $L$, $a$, and $b$ are known) Or does it not exist?
NullOverNull
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how to find the radius of convergence ? $\sum_{n=1}^{\infty} (\frac{x^n}{n} - \frac{x^{n+1}}{n+1})$

How can i find the radius of convergence ? i dont know where to start i cant use $\frac{a_n}{a_{n+1}}$ test here. $\sum_{n=1}^{\infty} (\frac{x^n}{n} - \frac{x^{n+1}}{n+1})$ the question looks simple but i dont know how to solve it i got that $R =…
Mather
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Intuitively, why is $\sum_{r=0}^n r^3 =\bigg(\sum_{r=0}^n r \bigg)^2$?

A duplicate question probably exists, but I couldn't find one. Please let me know if one is found and I'll delete this. We know that: $$\sum_{r=0}^n r=\frac12n(n+1)$$ and that: $$\sum_{r=0}^n r^3 =\frac14 n^2(n+1)^2$$ and these are easily proven…
Rhys Hughes
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Where's the error in this simple proof of this basic statement about summation?

Let $a_i$ be a sequence of real numbers and let p>1. Then: $\sum_{i \in\mathbb{N}} |a_i|\leq (\sum_{i \in\mathbb{N}} |a_i|^p)^{1/p}$. "Proof": $\sum_{i \in\mathbb{N}} |a_i|=((\sum_{i \in\mathbb{N}} |a_i|)^p)^{1/p}\leq (\sum_{i \in\mathbb{N}}…
Annie
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Computing $\sum_{i=1}^k{i\cdot c^i}$

I would like to compute the following sum : $$\sum_{i=1}^k{i\cdot c^i}$$ where $k$ and $c$ are any real numbers. I know how to compute $i$ and $c^i$ separately but I don't know how to do it when they are multiplied together. Thanks a lot for…
lou
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Sum of logs $\log x + \log\log x +...$

Is there a reduction for this infinite sum? $$\log x + \log\log x + \log\log\log x +... = ?$$ for all $x > 0$?
Matt Deleeuw
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Find The Integral Part Of The Series

Find the integral part of the series. $1/4^{1/3}$$+$$1/5^{1/3}$$+$$1/6^{1/3}$$...$$+$$1/1000000^{1/3}$. I tried to bring it down to a summation form, hence reducing it to $$\sum_{i=4}^n 1/i^{1/3}$$ But i still do not understand how to make it…
user636268
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Summa equation to solve

I'm having an hard time inverting this formula in order to get $x$. This is a short form for the amount of Energy deposited in a single pixel of a detector. $$ K = \sum_E f(E) e^{-(g(E) \cdot x)} $$ Where $f(E)$ and $g(E)$ are two functions summed…
suxdavide
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Simplify the following multiple summations

Could we solve the multiple summations $N = \sum\limits_{{j_1} = 1}^{K - \left( {q - 1} \right)z} {\sum\limits_{{j_2} = {j_1} + z}^{K - \left( {q - 2} \right)z} { \cdots \sum\limits_{{j_k} = {j_{k - 1}} + z}^{K - \left( {q - k} \right)z} \cdots …
Hui Zhao
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Index of summation - integer?

Can the index of summation of a sum be anything other than an integer? What is the reasoning behind the answer?
Christina Daniel
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How to show that these two summations are equal

$$\sum_{k=1}^{n-2} (n!-\frac{n!}{(n-k)!})\qquad (1)$$ $$\sum_{k=2}^{n-1} \frac{n!}{k!}(n-k)(k-1)\qquad (2)$$ I've got these summations as a solutions of a combinatorial problem. But I have no idea how to prove equality of them.
Mark Meister
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Finite sum on Negative Binomial

I am trying to work these out: $$\sum\limits_{u=0}^{k}{\left( \begin{matrix} r+u \\ u \\ \end{matrix} \right){{\left( {\alpha} \right)}^{u}}{{\left( {1-\alpha} \right)}^{r+1}}}$$ $$\sum\limits_{u=0}^{k}{\left( \begin{matrix} r+u \\ u …
Eln
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Closed form for finite summation of sequence $\sum^n_{i=1}{e^i/i}$

Does there exist a closed form for finite summation of the sequence $\sum^n_{i=1}{e^i/i}$ ?
Justin
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