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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What is the correct way to write sum from 47 to 54 so it means 404?

Does the above image correctly mean "This is a 404 error?" That is, does this $$ \sum_{i = 47}^{54} $$ mean 404? Or should it just be this, $$ \sum_{47}^{54} $$ without the equals?
Tim
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Help in explaining this sigma notation breakdown

I will appreciate some breakdown help which explains each step in the picture below to the last expression and the rules that applied to the changes. I am new to Sigma notations and thus confused.
MYSQLnoob
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Question about proof with sums

I am confused on how to write a formal proof for sum notations. How would I write a formal proof for this example? Prove that $$\sum\limits_{k = 0}^\infty\frac{2}{3^k} = 3.$$ Prove that for any $\alpha \in \{0, 2\}^\mathbb{N}$ that $$0 \le …
user165987
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Evaluating $\sum \sin(ak) x^k / k! $

Mathematica tells me that $$ \sum_{k=0}^\infty \sin(ak) \frac{x^k}{k!} = e^{x \cos (a)} \sin (x \sin (a)). $$ How would I go about evaluating such a series by hand? My first thought is to expand $\sin(ak)$, obtaining $$ \sum_{k=0}^\infty…
David Zhang
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How do i evaluate this sum?

Let $[x]$ be the nearest integer to $x$. (for $x=n+\frac{1}{2}, n \in N$, let $[x]=n$). Find the value of $$\displaystyle\sum_{m=1}^{\infty} [\sqrt m]^{-3}$$
user1001001
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How to solve $\frac{1}{n}\left[1+2\sum_{k=1}^{n-1}\frac{1}{\sqrt{\frac{n}{n-k}}}\right]$

I want to find an analytical expression for: $\frac{1}{n}\left[1+2\sum_{k=1}^{n-1}\frac{1}{\sqrt{\frac{n}{n-k}}}\right]$ I know that the result is independent of $n$ when $n$ is large, because I have used MATLAB for many different values of $n$, and…
hydrologist
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How find this sum $S(x)=\sum_{k=1}^{\infty}\frac{\cos{(2kx\pi)}}{k}$

Find this sum $$S(x)=\sum_{k=1}^{\infty}\dfrac{\cos{(2kx\pi)}}{k},x\in R$$ my idea: since $$S'(x)=2x\pi\cdot\sum_{k=1}^{\infty}\sin{(2kx\pi)}$$ then I can't.
math110
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How to find the sum: $1^{\frac{1}{3}}+2^{\frac{1}{3}}+3^{\frac{1}{3}}+ . . . +(50)^{\frac{1}{3}}$

Can some one help me to find the sum of the following expression? $$1^{\frac{1}{3}}+2^{\frac{1}{3}}+3^{\frac{1}{3}}+ . . . +(50)^{\frac{1}{3}}$$
Kumar
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Sum of all the positive integers with different digits till $100$

What is the sum of all the positive integers with 2 different digits till $100$ (without numbers with one digit and $100$) ? This was a problem I thought after hearing about Gauss and the Sum of integer till $100$. Here's how I tried to solve…
PunkZebra
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value of summation of $2^i\cdot i$

I'm trying to calculate the value of $$2^0\cdot0 + 2^1\cdot1 + 2^2\cdot2 + .... 2^n\cdot n$$ I figured this would be summation $2^i \cdot i$ from $i = 0$ to $n$. But iI'm unable to calculate its value. I have tried searching online but haven't…
otaku
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Is there a simplification of this sum of quadratic sums?

I have $a(1+2+3+...+N)^2+a(2+3+...+N)^2+a(3+..+N)^2+...+aN^2$, essentially a sum of quadratic terms holding an integer summation that get consecutively smaller Within each term the summation up to N elements (which could also go to infinity if that…
Majte
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Sum with conflicting specifier

This is the fomula (there might be some terms added, it is just boiled down to these two sums): $$\sum_{i = 0}^{N} \Big(\ Term_i * \sum_{j = 0}^{i - 1} x_j\ \Big)$$ The outer sum will iterate from $0$ to $N$ and the inner one from $0$ to $i - 1$. It…
Buni
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How prove this sum $\sum_{n=1}^{\infty}\frac{(-1)^n\ln{n}}{n^2}+\sum_{n=1}^{\infty}\frac{(-1)^n\ln^2{n}}{n}$

show that $$\sum_{n=1}^{\infty}\dfrac{(-1)^n\ln{n}}{n^2}+\sum_{n=1}^{\infty}\dfrac{(-1)^n\ln^2{n}}{n}=\dfrac{\pi^2}{12}\ln{2}+\ln^2{2}\left[\gamma-\dfrac{1}{3}\ln{2}\right]?$$ where the $\gamma$ is Eluer constant. I remember this sum is…
math110
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Understanding the summation and floor

I am beginner and a novice for sums and math in general. I don't know the steps or techniques to understand how does the right hand side equals the left? Could someone clarify the techniques or steps taken? Or direct me to a resource that…
enjoylife
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A sum that scales as the square root of the summands

Is there some (edit! analytic) expression $h(x)$ such that the sum $$\sum_{i=1}^n h(i)$$ scales as $O\left(n^\frac{1}{2}\right)$? Regarding the (40) comments under Sabyasachi's accepted answer: When you run a sum like…
Mantrissa
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