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 isolate variable in summation

Note: This question is actually for help writing a computer program, but the math is where I'm having difficulty, so I think it best to ask here. My goal is to transform an equation with this format (numbers chosen at random for example): Sum(x = 0…
BrianH
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$\frac{\sum^n_{i=0} x_i} {\sum^n_{i=0} x_i^2 } $

Hello I'm not sure how to evaluate this $$\frac{\sum^n_{i=0} x_i} {\sum^n_{i=0} x_i^2 } $$ is it $$\displaystyle \frac{x_0+x_1+...+x_n}{x_0^2+x_1^2+...+.x_n^2}$$ or $$\frac{1}{\sum^n_{i=0}x_i}$$ EDIT Now consider this $$\frac{\sum^n_{i=0}…
Elina
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Prove a finite sum equals to zero

I am trying to prove the following expression $$\sum_{n=1}^{N}\prod_{m=1, m \neq n}^{N} \frac{1}{\frac{1}{x_m}-\frac{1}{x_n}}=0. $$ (1) I have used an inductive approach to prove (1). In particular, when N= 2 or N=3, it is readily that (1) holds.…
PLe
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Need help solving the following summation

I need help solving the following summation: $$2^{x-1}\sum_{y=x}^{\infty} \frac y{2^y} $$ for $y\ge x\ge 1$ I tried writing out terms to solve but that did not go anywhere. I'm not sure what else I should try or if this fits a certain summation type…
the boy 88
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Inequality in summations

Let $$\sum_{k=0}^{n-1} \frac{nk^2-2k^3-7kn^2+10n^3}{n^4}=a_n$$ and $$\sum_{k=1}^n \frac{nk^2-2k^3-7kn^2+10n^3}{n^4}=b_n$$ then for $n=1,2,3,4,...$ then question is to find which among following is…
Navin
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Converting a double summation to a single summation

The following is an equation from my probability textbook: $\displaystyle\sum_y\sum_{\{x\ |\ g(x)=y\}}g(x)p_X(x)=\sum_x g(x)p_X(x)$ Could someone please give me a step-by-step explanation of why this is true?
Siddhartha
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Show that $\sum_{i=0}^{2n} (-1)^ix^i = \frac{x^{2n+1}+1}{x+1}$

How can one show that if $x\neq-1$: $$\sum_{i=0}^{2n} (-1)^ix^i = \frac{x^{2n+1}+1}{x+1}$$ I know that: $$\sum_{i=1}^{n} r^{i-1} = \frac{r^n-1}{r-1}$$ But then how to proceed?
GambitSquared
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Method of differences to find the sum of series

so I'm struggling with this one question or better any question of this type: You are given that $$\frac{3}{(5+3x)(2+3x)}\equiv \frac{1}{2+3x}-\frac{1}{5+3x}$$ use this result to find $$\sum_{r=1}^n \frac{1}{(5+3x)(2+3x)}$$ where n=100, giving your…
Elvendoork
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How does $\sum\limits_{i=1}^{\frac{n}{2}} \sum\limits_{j=i}^{n-i} \sum\limits_{k=1}^{j} 1$ equal $\frac{n^3}{8}$?

I'm trying to understand how to simplify summations. My text says that: $$\sum_{i=1}^{\frac{n}{2}} \sum_{j=i}^{n-i} \sum_{k=1}^{j} 1 = \frac{n^3}{8}$$ But does not explain how to get to the right-hand side. I think the above nested summation…
Larssend
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Compute the sum of an exponent divided by factorial

There was much discussion on Math SO why $$\lim_{n\to\infty} \frac{\alpha^n}{n!} = 0$$ when $\alpha > 1$. What would be the strategy for computing: $$\sum_{n=0}^{\infty} \frac{\alpha^n}{n!}$$ given that it's convergence is easlity proven using…
Denis Kulagin
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Why exchange second summation with (n-i)?

Pretty simple question. I always used this, because I know this holds, but to be honest I don't know why. $$ \sum_{i=1}^n \sum_{j:j
nummer92
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the summation of $-1^k(1\cdot3\cdot5\cdots(2k-1))x^{2k}$

I want to figure how and according to which formula is the summation $$\sum_{k=0}(-1)^k(1\cdot3\cdot5\cdots(2k-1))x^{2k}$$ according to which formula this equality is true
Tihraqua
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Derivation of formula for sum of first $n$ squares in Courant

In Courant's Differential and Integral Calculus (Vol. I), he presents the following derivation of a formula for the sum of the squares of the first $n$ integers: Link to Extract However, when I substitute $v=0,1,2,...n$ and sum all of the resulting…
A. Sable
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Difference of two sums

A sequence $a_n$ is given such that $a_n \rightarrow 0\wedge a_n>0\ \forall n$. I am to proof that $$\left|\sum_{n=1}^{\infty}(-1)^{n+1}a_n - \sum_{n=1}^{N}(-1)^{n+1}a_n\right| \le a_{N+1}.$$ I found out that…
Hendrra
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Summation of a series where n is odd

Find the sum of a series $$\ \rm 1^2+2(2)^2+3^2+2(4)^2+5^2.......+2(n-1)^2+n^2$$ where n is odd Attempt I tried $$\ \rm \sum^n_1 [2(n-1)^2+n^2] - 2\sum^\frac{n-1}{2}_1 [2(n-1)^2+n^2] $$ but the answer is coming wrong. The reasoning behind my…
mathnoob123
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