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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Find Sum for Equation

In my task, I've some questions to solve but I am stuck with the following one. Question: Find the following sum: $\frac{1}{22–1} + \frac{1}{42 –1} + \frac{1}{62 –1} + \dots + \frac{1}{202–1}.$ I tried to find many patterns but couldn't get one to…
Naila Akbar
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Nested summation and product notation of two expressions

Could someone please tell me how I should evaluate these two expressions together? I understand how each notation should be evaluated separately but not together. Also, would the way you evaluate change is the pi came before sigma? $$\sum_{i=1}^n…
Chenese
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Does the summation sign have to have lower values at the bottom?

I am working through a proof why determinants can be developed from any row or any column in a matrix. In passing, the authors claim that the summation $ \displaystyle \sum_{s=1}^{i-1}(\text{expression})$ has no values for $i=1$, since that would…
Magnus
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infinite sum of series with given recursive relation

Let $\{b_{n}\}$ be a sequence defined as $b_{n+1}=b^2_{n}-2$ and $b_{1}=b,$ where $b>2$. Then find sum of $$\sum^{\infty}_{n=1}\frac{1}{b_{1}b_{2}b_{3}\cdots b_{n}}=$$ What i try :: From equation $b_{n+1}=b^2_{n}-2$. Then $b_{2}=b^2-2$ And…
jacky
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How to Have and Modify and "Local Variable" in a Sum

Let's say I have polynomial $P(n,i)=(i+1)(n+1)^2$. I want to sum $k$ $P$'s, but with a twist: $n$ for any $P$ is based upon the $P$'s before it. So, in code, what I want is this: def P(n,i): return i*(n**2) V=0 S=0 for i in range(k): S+=P(V,i) …
DUO Labs
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Algorithm to represent $S$ as sum of $K$ numbers between $1$ and $N - 1$

Given $N$, $K$ and $S$, we have to find out a way to represent $S$ as a sum of exactly $K$ numbers between $1$ and $N - 1$. For example, if $S = 22$, $K = 7$ and $N = 10$, then one possible way is $3 + 3 + 3 + 3 + 3 + 3 + 4 = 22$. Is there any way…
Robur_131
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Finding a closed formula for a sum

I wrote this sum (out of the blue) and wondered if it has a closed form: $$\sum_{k=1}^{\infty} L^{\frac{1}{k}} \cdot(-1)^{k+1}$$ where $L \in \mathbb{N}$ I thought of a sum that would use "$\text{k-root}$" but with alternating sign ($+$ to $-$…
CSch of x
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Calculate a finite sum

I want to find out the best way to perform a concret summation. I know for instance that: $$ \sum_{i=1}^{p}\left \lfloor \sqrt{i} \right \rfloor=\int_{1}^{p+1}\left \lfloor \sqrt{i} \right \rfloor di $$ and then $$ \int_{1}^{p+1}\left \lfloor…
Josi
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Proving a series is less than an expression

In the book I am reading I am asked to prove the following: $$\sum_{n=1}^k{\cos n}<\frac{1}{2\sin{\frac{1}{2}}}-\frac{1}{2}$$ My attempt: $$\sum_{n=1}^k{\cos n}=\frac{1}{2\sin{(1/2)}}\sum_{n=1}^k{2\sin{(1/2)}\cos…
Samuel A. Morales
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Simplification of $\sum_{\substack{G\in\{0,1\}^n\\G_i=1}} (-1)^{G^\top\vec{1}}\frac{\lambda_i}{G^\top\lambda}.$

Let $\lambda\in\mathbb{R}^n$ be a vector with positive components. Denote by $\vec{1}\in\mathbb{R}^n$ the vector with each component equal to $1$. I am seeking to simplify the following sum for any fixed…
Tullio
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sum of the first $n$ terms of a quadratic sequence

$$\sum_{i=0}^n(2i+3)^2=\frac{(n+1)(4n^2+20n+27)}{3}$$ Hey! :) I need help with this problem, I'm kinda silly and I've got a problem with this, I don't even know how to start. If you guys can help me that would be amazing. I'm in my first year of…
pablo.
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How is $n\sum^n_{j=1}(x_j-\overline{x})^2$ derived?

Could somebody explain how they derive the term in the red highlight. What I get: $$n\sum^n_{j=1}(x_j)^2-n^2\overline{x}^2=n\left(\sum x_j^2-n\overline{x}^2\right)$$ $$=n\left(\sum x^2_j-\frac{n}{n^2}\left(\sum x_j\right)^2\right)=n\left(\sum…
user634512
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Is there a solution to this ? $f_0(n)=2n−1,\\ f_{k+1}(n)=\sum_{i=1}^nf_k(i)$

after developing a formula, I came up against this : $$f_0(n)=2n−1,\\ f_{k+1}(n)=\sum_{i=1}^nf_k(i)$$ So, for example :$$f_1(n)= n^2\\ f_2(n) = \frac{n(n+1)(2n+1)}{6}$$ Can we find $f_n(n)$ ? Thanks to Jens Renders for his advice of rewriting it…
Theman Op
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How do we know the range of a double summation?

In this snippet from Knuth's Concrete Mathematics, I'm not sure how we would know the range of $j, k$: On the left hand side of the above equation, we just are given that $1 \leq j, k \leq 3$ then how do we know that $1 \leq j \leq 3, 1 \leq k \leq…
Katie Melosto
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Decimals in sigma notation

What would happen if you input a decimal into sigma notation like this: $$\sum_{n=0.5}^2n^2$$ Would this be $0.5^2+1^2+1.5^2+2^2$?
qianqian07
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