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.

17770 questions
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Summation notation with a comma- what does this mean

I am reading a document with this expression: evidencei=∑jWi, jxj+bi Can someone explain what this means? In context: We also add some extra evidence called a bias. Basically, we want to be able to say that some things are more likely independent…
Thomas Shera
  • 143
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How to pove this identity with sum

show that $$\sum_{k=1}^{n}\left(\binom{n}{k}\dfrac{\Gamma{(k+1)}}{\Gamma{(k+1-a)}}x^{k-a}\right)=\dfrac{\Gamma{(n+1)}}{\Gamma{(n+1-a)}}(1+x)^{n-a}$$
user223800
3
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2 answers

Double Summation Simplification (Simple)

My questions: For the inner summation, When $j=i$ does that mean $j$ will equal $2$ because $i$ equals $2$? I'm confused and stuck on the yellow, because I thought it would just be $n$. NOT ($(n-i+n)$. I understandstand Line 1 of the problem…
Kapooky Handy
  • 595
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6 answers

Help with summation: $\sum_{k=1}^\infty\frac{k(k+2)}{15^k}$

How can one evaluate the below sum? Any help would be greatly appreciated. $$\sum_{k=1}^\infty\frac{k(k+2)}{15^k}$$
Panglossian Oporopolist
  • 5,223
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Deriving the formula for the $n^{th}$ tetrahedral number

$T(n)$ is $n^{th}$ triangular number, where $T\left(n\right)=\frac{n^2+n}{2}$ And from other sources I know the $n^{th}$ tetrahedral number is $G\left(x\right)=\sum _{n=1}^xT\left(n\right)=\frac{\left(x^2+x\right)\left(x+2\right)}{6}$ I also…
theideasmith
  • 1,090
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Solve summation for variable that is in upper limit and summation

I don't remember how to solve these, but I hope someone can refresh my memory. I have the following equation, how can I solve for t? $$50000 = \sum_{i=0}^{12t} 1.004^{12(t - i/12)}$$ EDIT: if it helps at all, expands out to $$50000 = 1.04^{12t} +…
holtc
  • 133
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Formula for the sum of $\ n\cdot 1 + (n-1)\cdot 2 + ... + 2 \cdot (n-1) + 1\cdot n$

I'm looking for a simpler formula to calculate this sum: $$n\cdot 1 + (n-1)\cdot 2 + ... + 2 \cdot (n-1) + 1\cdot n$$ Alternate representation (but should be equal to the above): $$\sum \limits_{k=1}^{n}(n+1-k)\cdot k$$ Rationale behind requested…
holroy
  • 133
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Value of double sum of powers of fractions between 0 and 1

Is there any way to find closed form for the sum (where k is positive integer) $$S = \sum_{i = 1}^{n}\sum_{j = 0}^{i} \left( \frac{j}{i} \right) ^ k$$ Using Faulhaber's formula I got $$S = \frac{1}{k + 1} \sum_{r = 1}^{k + 1} (-1) ^ {\delta_{k, r}}…
Reactant
  • 677
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Find $S=\sum\sum\sum x_{i}x_{j}x_{k}$ where $x_{i}=-x_{n-i+1}$ for $1\leq i\leq n$

Suppose that $x_{1},x_{2}.....x_{n},(n>2)$ are real numbers such that $x_{i}=-x_{n-i+1}$ for $1\leq i\leq n$. Consider the sum $S=\sum\sum\sum x_{i}x_{j}x_{k}$, where the summation are taken over all $i,j,k: 1\leq i,j,k\leq n$ and $i,j,k$ all are…
Akshay Hegde
  • 757
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2 answers

Summation of product of $m+1$ alternate numbers

We know that $$\begin{align} \sum_{r=1}^n \prod_{j=0}^m (r+j)&=\sum_{r=1}^n r(r+1)(r+2)\cdots(r+m)\\ &=(m+1)!\sum_{r=1}^n\binom {r+m}{m+1}\\ &=(m+1)!\binom {n+m+1}{m+2}\\ &=\frac {(n+m+1)^\underline{m+2}}{m+2}\end{align}$$ but is there a similar…
Hypergeometricx
  • 22,657
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How can I simplify $1\times 2 + 2 \times 3 + .. + (n-1) \times n$ progression?

I have a progression that goes like this: $$1\times 2 + 2 \times 3 + .. + (n-1) \times n$$ Is there a way I can simplify it?
Max Yankov
  • 155
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How to write product as a nested sums

I'm required to write $\prod\limits_{i=1}^n(1-j_{i})$ as a nested sum, where $j_{ll} \neq j_{k}$ if $u \neq k$. I undestand I'd get something in the form of $1-\sum\limits_{i=1}^n{j_{i}}+...+(-1)^n\prod\limits_{i=1}^nj_{i}$ but I don't know how I…
gpr1
  • 491
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1 answer

What's this problem getting at? (sum and derivative)

Let $$P(x) = \sum_{m=1}^n b_m (x-a)^m $$ Then $$P^{(k)}(x) = \sum_{m=1} ^n b_m(m)(m-1)\cdots(m-k+1)(x-a)^{m-k}$$ Is above correct? But what about when $m < k$, that seems to ruin the formula? Then, a side question to above assignment is posed, and…
Commour Touess
  • 31
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Is $\sum_{n=1}^\infty \frac{\left(-1\right)^{n+1}\log\left(n\right)}{n}$ divergent?

Is \begin{align}\sum_{n=1}^\infty \frac{\left(-1\right)^{n+1}\log\left(n\right)}{n}\tag{1}\end{align} divergent? I think it is because by comparison \begin{align} \sum_{n=1}^\infty \frac{1}{n}<\sum_{n=1}^\infty…
bjd2385
  • 3,017
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reverse of a summation formula

This is the formula: $\sum_{i=1}^n x^i = a$ For example: $\sum_{i=1}^4 3^i = 120.$ If we have x and a, is it possible to obtain n? ps: I'm sorry if it's a silly question!
ali
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