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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How to interchange the order of summation in the triple summation

I encounter following problem i have triple summation where limit of inner summation depends on indexes of previous two summation (which commutes) $\sum_{i=0}^a\sum_{j=0}^b\sum_{k=0}^{j+i}$ and a desired form is to change order of summations to move…
Tupta
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Calculating $\;\sum_{k=0}^{\infty}\sqrt k\cos(kx)$

I am curious if there is a closed form for this summation $\sum\limits_{k=0}^{\infty}\sqrt k\cos(kx)$ I am aware that $\sum\limits_{k=0}^{\infty}\cos(kx)$ resembles a dirac delta comb and $\sum\limits_{k=0}^{\infty}k\cos(kx)$ can be expressed as…
Srini
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Sum with Multi-Index Notation

I have a doubt about sums with multi-index notation, maybe its pretty elementary but I need to clarify it: If $\alpha=(\alpha_1, \ldots, \alpha_n)\in\mathbb N^n\cup\{0\}$ can I write $$\sum_{j_1=0}^{\alpha_1}\cdots…
PtF
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Proof of a double summation reduction identity

I'd like to see a proof of this identity I discovered: $$ \sum_{i=1}^{n}\sum_{j=1}^{i}a_j = \sum_{i=1}^{n}a_i(n-i+1) $$ Probably there's something I forgot because I couldn't manage it. [edit]Possibly with just identities, not examples…
user2488812
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how do you get the following summation.

$\sum_{r=0}^n \frac{(-1)^{r-1}\binom{n}{r}(1-x)^r}{r}$ I tried it using $\binom{n}{r}$ = $\binom{n-1}{r}$+$\binom{n-1}{r-1}$ but I am not getting the desired result. given answer is $\frac{1-x}{1}$+$\frac{1-x^2}{2}$…
Baibhaw Kumar
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Simplify $\sum^{N-1}_{k=1} \frac{1}{1-\exp(i2\pi k/N)}$

I am trying to prove that $$\sum^{N-1}_{k=1} \frac{1}{1-e^{i2\pi k/N}}=\frac{N-1}{2},$$ the closest formula I can think of is $$\sum^{N-1}_{k=0} r^k = \frac{1-r^N}{1-r},$$ but seems like it is not exactly the case.
Marcy Cheng
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Are these summation notation the same or do they express different rules?

Are these two the same? $\sum_{u\in S}\sum_{v\in S}f(u,v)$ $\sum_{u,v\in S}f(u,v)$
Piskator
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What are the steps to get the right side of this summation equation?

I am computing the time-complexity of an algorithm and came up with the following summation. I plugged it in to Wolfram Alpha to see what the result is. Unfortunately, WA does not show the steps of this particular solution. Can someone show me how…
Amjad
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Triple summation having $i,j,k$ as variables

Finding value of $(1)\ \displaystyle \mathop{\sum\sum\sum}_{1\leq i
Priti Bisht
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Is the meaning of a formula using the summation symbol followed by a multiplication and an addition ambigous?

I have come accross the following formula: $ y = \sum\limits_{i=1}^{n} w_i \cdot x_i + b $ The idea that the author wanted to express was to add all products $w_1 \cdot x_1 + w_2 \cdot x_2 \ldots $ and then to add $ b $ once. However, without…
René Nyffenegger
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Proving a Sum Without Induction

I wanted to find the sum $\displaystyle\sum_{k=1}^n\frac{1}{k^2+k}.$ I have proven using induction that the sum is $\frac{n}{n+1}$, but I was wondering if there was another method that didn't use induction.
AreS
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How to expand infinite sum expressions.

I saw this in a paper (however, I cannot derive it): We define the growth rate of consumption as $g_{C, s} \equiv \frac{C_s}{C_{s-1}}$ and use $C_t=\left(\prod_{s=1}^t g_{C, s}\right)C_0$ to rewrite lifetime utility, $\mathcal{V}_0=$ $\mathbb{E}_0…
Emmanuel Ameyaw
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How to show that the summation equal to $(t(n)+1)$

Why is $$ \sum_{n=1}^{\infty} x^{2n^2} +\sum_{n=1}^{\infty} x^{8n^2} + \sum_{n=1}^{\infty} x^{32n^2}+\cdots = \sum_{n=1}^{\infty}(t(n)+1) x^{2n^2} $$ Where $t(n)$ is the highest power of 2 which divides $n$ My attempt : list $t(n)$ first and make…
Fadhlan Zhaahiran
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