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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Showing a compact form of a summation.

How can I show that: $$P^k \left(1 + \frac {2}{p} + \frac {3}{p^2} + \frac {4}{p^3} + ..... + \frac{k}{p^{k-1}} + \frac{k+1}{p^k}\right) = \sum_{i = 0}^{k} (k + 1 -i) p^i$$ Could anyone explain this for me please?
Intuition
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Symbolic evaluation of the double series with terms $1/[a^2 + (b + m)^2 + (c+n)^2]$

Any chance to sum this series? ($a,b,c$ are real) Sum[((a^2 + (b + m)^2) + (c + n)^2)^(-1), {n, -∞, ∞}, {m, -∞, ∞}] Or even a more general one? Sum[((a^2 + (b + d m)^2) + (c + e n)^2)^(-1), {n, -∞, ∞}, {m, -∞, ∞}] I don't know if it can help, but…
wondering
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Calculating the infinite sum $1-\frac 1 7+\frac 1 9 - \frac{1}{15} + \frac 1 {17}\mp ...=\frac{1+\sqrt{2}}{8}\pi$

Prove that $$1-\dfrac 1 7+\dfrac 1 9 - \dfrac{1}{15} + \dfrac 1 {17}\mp ...=\dfrac{1+\sqrt{2}}{8}\pi$$ My attempt: I tried to break it into two series $$(1+1/9+1/17+...)-(1/7+1/15+1/23+...)$$ But I don't know how to proceed. Any hints would be…
Pranav Aggarwal
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Simplification of $\sum \limits_{a=0}^{m} \sum \limits_{b=0}^{n} x^{|a-b|}$

I was trying to determine a simplification of $f(x) = \sum \limits_{a=0}^{m} \sum \limits_{b=0}^{n} x^{|a-b|}$. One approach I considered was using a case bash, but this was quite inelegant so I tried to find a more elegant method. Another method…
Sharky Kesa
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Check on Summations

I'm completing some time complexity analysis on loops which requires me to represent psuedocode loops as summations. I have two summations that I believe I have done to a degree of correctness, but am unsure. Here's the initial pseudocode for the…
bpryan
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$\sum_{n=1}^{\infty}\sum_{i=nr}^{\infty}f(i)=\sum_{i=r}^{\infty}\sum_{n=1}^{\left\lfloor \frac{i}{r}\right\rfloor}f(i)$

This was given as part of the answer of a more complex problem: $$\sum_{n=1}^{\infty}\sum_{i=nr}^{\infty}f(i)=\sum_{i=r}^{\infty}\sum_{n=1}^{\left\lfloor \frac{i}{r}\right\rfloor}f(i)=\sum_{i=r}^{\infty}\left\lfloor\frac{i}{r}\right\rfloor…
ficoteh
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Closed form expression of an exponential sum from -n to n

I'm looking for a way to write the following sum as a closed form expression: $$ \sum_{n=-N}^{N} e^{cn*i} $$ where $c$ is a constant and $i$ is the imaginary unit. I found a closed form solution for $n= [0, N-1]$ at…
Skipher
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Changing the order of summation in a triple sum

I am trying to simplify an expression that came up when I was trying to calculate the energy of a multiple pendulum system. The expression I have is $$\sum_{k=1}^N \sum_{i=1}^k \sum_{j=1}^k m_k v_i v_j \cos(\theta_i-\theta_j)$$ but I want to change…
LeoSporm
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How to find $a$ & $b$ that satisfy $10^{n+1}=\sum_{i=1}^{n}{(a_{i}10^{i}-b_{i}9^{i})}$?

I have an equation of the form: $$10^{n+1}=\sum_{i=1}^{n}{(a_{i}10^{i}-b_{i}9^{i})}$$ where $0 < a_{i} \leq 20$ and $0 < b_{i} \leq 18$. By inspection, I can see that for $n=1$, $a=19$ and $b=10$ is a solution and I can verify in a spreadsheet that…
Rasputin
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How to simplify $ \frac { \sum_{n=0}^{ \infty } \frac {(-1)^{n-1}n^{n-1}x^{n}} {n!}} {x\sum_{n=0}^{ \infty }\frac{x^n}{n!} }$?

How can I simplify $ \frac { \sum_{n=0}^{ \infty } \frac {(-1)^{n-1}n^{n-1}x^{n}} {n!}} {x\sum_{n=0}^{ \infty }\frac{x^n}{n!} }$ so that I can evaluate it?
user604496
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Summation of 'for loop' with conditional?

I am trying to convert instances of nested 'for' loops into a summation expression. The current code fragment I have is: for i = 1 to n: for j = 1 to n: if (i*j >= n): for k = 1 to n: sum++ …
bpryan
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How do I get the equality $\sum_{l-s}^{l+s}(2j+1) = (2l+1)(2s+1)?$

I am having trouble evaluating $$\sum_{l-s}^{l+s}(2j+1)$$ I checked it for $l=3$, and $s=1$, but I am looking for the general solution. My notes tell me that the expression should be equal to $$(2l+1)(2s+1)$$ but I don't see how that came about. (I…
zabop
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Reformulating a sum

I´m struggling to disentangle this sum... Stuck in getting from line 2 to 3. Could someone give me a hint what the rationale behind this is? In particular, why are we squaring inside the summation here? $$\sum_{t=2}^\tau G_t=\sum_{t=2}^\tau…
Alfi
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floor value of a function involving factorial sum

Evaluation of floor value of $(2019!)\cdot (2018!+2017!+\cdots +2!+1!)^{-1}$ Try: I am trying to solve it using gamma function. $\displaystyle (n-1)!=\Gamma (n)=\int^{\infty}_{0}e^{-x}\cdot x^{n-1}dx$ and $\displaystyle…
DXT
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Recursive summation of summations

So basically what i'm trying to find is: -Start with $\sum_{i=1}^j 1$, which obviously is $j$ -Do the summation of the answer, which is: $\sum_{j=1}^k j$, which is $k*(k+1)/2$ -Next step is the summation of all of this: $\sum_{k=1}^l…
Miquel Martnez de Morentin
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