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 find the summation of the above series up to infinity?

How to find the summation of the above series up to infinity? I have tried to find the above summation by putting n=0,1,2,3,...and so on. Finally I got the above series to be (1-(1/2)+(1/23)-(1/24)+(1/45)-(1/46)+...) and so on. But I don't know how…
DEEPANWITA Roy
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Merging a double summation into a single one

I'm looking for merging this summations into a single one: $\displaystyle \sum_{i=0}^k{\sum_{j=0}^i{b^j}}$ where $b$ is an integer. I know it is equal to: $\displaystyle \sum_{i=0}^k{b^i(k-i+1)}$ In fact, one can see that: $\displaystyle…
Lava
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Prove $\sum_{x=1}^{X} \sum_{y=1}^Y {Y \choose y} \times (x-1)^{Y-y} = X^Y$

I accidentally find this property, $\sum_{x=1}^{X} \sum_{y=1}^Y {Y \choose y} \times (x-1)^{Y-y} = X^Y$, when doing a brain-teaser. I only tried up to y=3, but it seems to be true for all X and Y. Can anyone give an induction prove? (or any direct…
Gabriella Chaos
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Why is $\sum_{k=0}^{n-1} (-1)^k a^{n-1-k}b^k = \left(\sum_{k=0}^{n-2} (-1)^k(k+1)a^{n-2-k}b^k\right)(a+b)+(-1)^{n-1}nb^{n-1}$?

This transformation was used in answers to these questions: $\gcd\left(a+b,\frac{a^p+b^p}{a+b}\right)=1$, or $p$ Proving $\gcd( m,n)$=1 (the name is weirdly formulated, but those two are the same questions). To someone who already asked the question…
anie
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Gauss' method of summing from 1 to 100

This may be a rather silly question, but I found myself thinking about the famous anecdote of Gauss summing from 1 to 100 in school, where he sums $$1+100=101, 2+99=101,...$$, and so on. He concludes that there are 50 pairs of numbers between 1 and…
Lorentz
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Evaluating trig sums

How can one approach the evaluation of these sums? I have attempted to expand them using trigonometric identities, but I am unable to discern a viable pattern. $$S1= \sin x \cos 2y + \sin 2x \cos 3y +... + \sin (n-1)x \cos ny$$ $$S2 = \cos x \sin 2y…
Kiiza Bandekye
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Does this sum have an analytical formula?

I'm attempting to derive an analytical formula for the following sum: \begin{align} \sum_{j=1}^k \sin(\theta j)\sqrt{\sin^2(\theta j)+a}, \end{align} where $a \geq 0$. If anyone has any insights on how to find such a formula, or if it is not…
Joe
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Basic summation issue

I am having some trouble prooving the following summation formula. As a and b are constants I got the constant part out of the sum and tried to split the summation (such that sum of a to b is the same as sum of 1 to b minus sum of 1 to a plus a for…
evilmandarine
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Equating terms of a summation

Suppose I have an equation of summations in the form: $$\sum_{k=0}^{\infty}a_kz^k=\sum_{k=0}^{\infty}b_kz^k$$ Under what conditions and how can I deduce that: $$a_k=b_k\ \forall k$$ Thank you in advance for your answer
Bosnan
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How to write the sum of r^(α+1) in terms of the sum of r^(α-1)?

The sum from 1 to n of r^(α+1), {α,n} $\in ℕ $, can be written as the sum from 1 to n of the sum from x to n of r^α. $$\underset{r = 1}{\overset{n}{\sum }}r^{\alpha + 1} = \underset{x = 1}{\overset{n}{\sum }}\underset{r = x}{\overset{n}{\sum…
Ignorabimus_Sol
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How to calculate the sum

Is there any easy way to calculate $\sum_{a=0}^m \sum_{b=0}^m \sum_{c=0}^m \min_2(a,b,c)$ as a function of $m$ ? $\min_2(a,b,c)$ is the second minimum of $a,b,c$. That is if $a \leq b \leq c$, $\min_2(a,b,c)=b$. $m$ is a positive integer.…
user12290
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Why does this sum equal zero for arbitrary n?

In the middle of a physics calculation I have encountered a sum of the form: $$\sum_{j=1}^n\bigg(\sum_{l=1\\ l\neq j}^n\dfrac{1}{x_j-x_l}\bigg)^2$$ The way the book proceeds implies that the sum of the terms which involve a product of two different…
Wild Feather
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How does the following summation evaluate to $n-i-1$?

I have the following summation: $$ \sum_{j=i+1}^{n-1}1=n-i-1 $$ I'm trying to understand how the summation evaluates to the expression $n-(i+1)$. Can someone show me algebraically how this is correct?
Trevor Judd
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Sum from $1$ to $(N-1)/2$

On a piece of coursework I am working on, we are asked to perform some calculations involving the following: As you can see, it is possible for $(N-1)/2$ to be fractional, e.g. in the case $N=4$. How exactly do I go about computing this sum, as I…
TerryStone
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Unweighted sum from weighted sum?

Suppose that we have $\frac{\sum_{i=1}^{m}x_i\ln{w_i}}{\sum_{i=1}^{m}\ln{w_i}} < C$ with $x_i > 1$ and $\ln{w_i} > 1$ for all $1 \le i \le m$, $x_i$ being integers, and $w_i$ and $C$ being rational numbers. Is it possible for me to know the minimum…
k1234567890y
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