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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Is this equation sum setup right passing the N an i values?

So I have been working on this equation which sums up two summations from i = 1 to N. This is the equation. This is the latex equation for it for help in editing if its wrong. AL = \sum_{i=1}^{N}\left ( \sum_{i=1}^{N}max(0, m + d(r_{a},r_{p}) -…
MNM
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Partial Sum of $\sum^{n-1}_{i=0} \frac{n-i-1}{n-i}$

I am looking to find the partial sum of the following sum for an algorithm I am analyzing: $\sum^{n-1}_{i=0} \frac{n-i-1}{n-i}$. I can see it can be simplified to: $\sum^{n-1}_{i=0} \frac{n-i-1}{n-i}$ $= \sum^{n-1}_{i=0}…
yosmo78
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Not able to understand the notation used in the summation.

In the specified equation, the lower limit of the inner summation ('$i$' with numerical subscript) seems to have all the values as same, in each loop of summation, which it clearly can't be. I'm not able to understand how to expand this summation…
Justin K
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Double summation swap

the solution sheet of an exercice I was working on says in particular: $$ \sum_{k=0}^{n}\sum_{i=0}^{k}\binom{k}{i}t^ia^{k-i}\alpha_k = \sum_{i=0}^{n}t^i\sum_{k=i}^{n}\binom{k}{i}a^{k-i}\alpha_k $$ I can convince myself of this by spelling out…
Mogu
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Solving A Double Summation

I came across this problem in my textbook: $\sum_{i=10}^{n+10}\sum_{j=i}^{n+10} j$, I know how to do a double summation but not one where they are linked at the starting points like this. The textbook says to write this summation as a function of…
Bob Joe
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Compute $\sum_{n=1}^{1010} (-1)^{n+1} \binom{2020-n}{n-1} 2^{2021-2n}.$

Compute $$\sum_{n=1}^{1010} (-1)^{n+1} \binom{2020-n}{n-1} 2^{2021-2n}.$$ Everything is in the summation I wanted to test out the first few values but then $2^{2021-2n}$ gets too large. One thing i noticed was that i could take out the 1 and -1 on…
user829751
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Process to Sum Outputs of a Function

I was wondering if there was any process to take the sum of any given functions outputs akin to a integrals to sum the area under the curve. For example, if my function is $$ \frac { \left( x - \frac 1 2 \right) ^ 2 } 2 \text , $$ and I want to find…
AaronSzcz
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If Bounds Don't Match Expression in Summation

If I have: $$\sum_{k=1}^n \sum_{i=1}^k(2k)$$ Why does the inner sum become $k$? I can see that it is you can "pull out" the $2k$ and it would leave a $1$, but why can we pull this out? Is it because the lower bound doesn't match the bound in the…
user750949
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Inequality involving sums of powers

I am wondering how to prove (1) $b + b^2 + ... + b^{n - 1} < b^n$ for all $b, n \in \{2, 3, ...\}$ and more generaly (2) $a_1b + a_2b^2 + ... + a^{n - 1}b^{n - 1} < b^n$ when $a_i \in \{0, 1, .., b - 1\}, (a_1 \neq 0)$. I am studying number systems…
1b3b
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How can simplify this summation notation

So I have just started learning about summations, and I am stuck with this problem for a while, I need to simplify this summation to get a simpler expression, $\sum_{i=1}^n i(i+1)$ where n is given. Any help would be much appreciated and Thank you.
Peter
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Solve for variable that is the upper limit of a summation

It has been a really long time since I had to solve a problem like this and I was wondering if I could get some assistance with it. $$x= \sum_{i=1}^{t} \frac{1}{i^c} \hspace{.25cm} \text{where } c \geq 0 $$ I am trying to solve for t for a given x. …
Jedi_Maseter_Sam
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Calculating Nested Sums

I know that $$ \sum^{n}_{k_p=0} \sum_{k_{p-1}=0}^{k_p} ...\sum_{k_1=0}^{k_2}1 = {n+p\choose n} $$ Now I would like to calculate the closed-form solution for the following $$ \sum^{n}_{k_p=1} \sum_{k_{p-1}=1}^{k_p} ...\sum_{k_1=1}^{k_2}x_{k_1} =…
Esildor
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Average number of comparisons made by the Binary search (summation computation)

I am trying to find out the average number of comparisons made by the Binary search. In that, I'm confronted with following summation :- Avg = $(1/n)\sum_{i=1}^{\log_2 n} i*2^i$ In this summation, $i$ are the number of comparisons made at each node…
Onkar N Mahajan
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Is there any efficient way to evaluate the following summation?

Given $r_1,\ldots,r_n\in\mathbb{R}$, the objective is to compute the following summation: \begin{align} \sum_{\substack{x_1,\ldots,x_n\in \{0,1\}\\x_1+\ldots+x_n
Heedong Do
  • 339
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Write the following series in sigma notation: $3 + 10 + 21 + 36 + 55$

What I need is an explanation of the method. I tried to get the general formula (for the nth term) of the series but I can't find it.
Inmaculada Collados
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