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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Elementary proof of $\sum_{k=0}^\infty \frac{1}{(2k+1)^2-(2n)^2}=0$

Using complex analysis, it is well known, that $$f(s)=\sum_{k=1}^\infty \frac{2s^2}{k^2+s^2} = \pi s \coth(\pi s) - 1 \, .$$ From this it is clear, that $$f\left(i(n+1/2)\right) = \sum_{k=1}^\infty \frac{-2(2n+1)^2}{(2k)^2-(2n+1)^2} = -1 \, .…
Diger
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Closed form for $\sum_{k=0}^{n-1} \sin(m\theta_k)(\pi-\theta_k)$

In the process to compute $\int_0^{+\infty} \frac{t^{m-1}}{1+t^{2n}}{\rm d}t$ (see this thread), I got stuck on this sum : $$S_{m,n} = \sum_{k=0}^{n-1} \sin(m\theta_k)(\pi-\theta_k)$$ where $m$ and $n$ are integers, $1\le m<2n$, and…
Nicolas FRANCOIS
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Normalize sum of values to percentages from 0-100

I have two values that sum to one in an arbitrary manner: [.6 , .4] [-.2, 1.2] [1.9, -.9] etc... How can I "normalize" these values so that they represent a percentage from 0-100%? What mathematical concepts should I be looking at to make this…
ph34r
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Find list of N increasing values that add up to S given start value V

I have a list of N items The price of the 1st item in the list is valueA I want the price to gradually increase from one item to the next one in the list. If I sell all N items, I want my total earnings to be T. I'm looking for the price each item…
Sprout Coder
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formula for finding an upper limit for summation

I am summing over percentages list Per = [5,6,1,4] this list has 4 elemets so I am summing as follows : $\sum_{i=1}^{4} per(i)$ the sum is 16 and the half of this sum is 8 $1/2\sum_{i=1}^{4} per(i)$ now I want to sum over the same list to find a k…
Cav
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Find $c_n$ s.t. $\sum_{n=1}^{N} \sum_{k=0}^{n} a_{n,k} x^{n+k}=\sum_{n=1}^{2N} c_n x^n$

For arbitrary coefficients $a_{n,k}$, I am attempting to group by equivalent powers of $x$. $$\sum_{n=1}^{N} \sum_{k=0}^{n} a_{n,k} x^{n+k}=\sum_{n=1}^{2N} c_n x^n$$ I wish to express $c_n$ as a sum of finitely many $a_{n,k}$. Thank you all very…
Talmsmen
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Sigma Notation to Discrete Values

I was reading the top voted answer for What is the average of rolling two dice and only taking the value of the higher dice roll? I ran into the…
Chris Rudd
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Summing a quadratic series

I'm creating a game, the exp required for a particular level is: (level + 3) ^ 2 * 10 Total exp required for level 1 = (1 + 3) ^ 2 * 10 level 2 = (1 + 3) ^ 2 * 10 + (2 + 3) ^ 2 * 10 level 3 = (1 + 3) ^ 2 * 10 + (2 + 3) ^ 2 * 10 + (3 + 3) ^ 2 *…
HL.
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Summation of three dependent variables with a given constraint

Find the minimum and maximum value of $\sum_{i,j,k} ({x_{ij} + x_{ik} + x_{jk}})$ given that $\sum_{i,j} {x_{ij}} = 225$ and $1\leq i
Orion_Pax
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Biggest product from combination of natural numbers

Given a natural number N, find the combination of natural numbers that sums N and has the biggest product. For example: 8 gives 3x3x2 = 18 10 gives 3x3x4 = 36 20 gives 3x3x3x3x3x3x2 = 4374 I "guess" that the answear is: divide the number by…
Wolfgang Amadeus
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How can I compute this sum of a series?

After some long derivations with a certain 1D Hamiltonian, I end up with the following sum $$\sum_{n=2}^\infty \frac{1}{e^{an^2(1+\frac{b}{4n^2-1})}-1}$$ But I am completely stumped by the $(1+\frac{b}{4n^2-1})$ term. Is there any trick I can use to…
DarkBulle
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Volume of pyramid made from cubes

I'm trying to calculate the volume of a pyramid that is made from a descending number of cubes per layer for example: the bottom layer is 4x4, the second layer is 3x3, the third layer is 2x2, and so on one boundary I am facing is that I only know…
an inconspicuous semicolon
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Greatest integer that is less than or equal to $\sum_{n=1}^{100}\frac{1}{\sqrt{n}}$

Basically I want to find the GIF of $\sum_{n=1}^{100}\frac{1}{\sqrt{n}}$ I have no idea how to do it. Can anyone give me a hint for solving this. thanks!
Shlok Jain
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How can I show that each part of the two sums are equal if the sums are equal?

For some $x \in N$ and $n \in N$, suppose $0 \leq a_i < x$ and $0 \leq b_i < x$ for $i=0,...,n-1$ and that the following is true $$ \sum_{i=0}^{n}a_i\cdot x^i = \sum_{i=0}^{n}b_i\cdot x^i $$ How can we show that $a_i\cdot x^i$ = $b_i\cdot x^i$ for…
David Callanan
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Find summation formula

$$\sum_{m=p}^t \left ( \frac{x}{100 \cdot m} \right )$$ Right now I have this, $x$ is fixed(it will never change), the function is $f(n) = \frac{x}{100n}$ so its summation from $n=p$ to $n=t$, where the formula would be $f(n)$, how can I find this…
Pedro Paulo Monte Pagani
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