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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Closed form for sum of constant to the power of a quadratic polynomial in the index variable.

How do I find a closed form solution of the following sum? $$\sigma=\sum_{i=1}^n \gamma^{\alpha i^2 + \beta i}=\sum_{i=1}^n \gamma^{\alpha i^2}\gamma^{\beta i}$$ My attempt: I tried using the method which would solve $\sum\gamma^{\alpha i}$…
Alice Ryhl
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Simplifying the sum $\sum_{i=1}^n \sum_{j=i+1}^n({(x_i-x_j)}^2+{(y_i-y_j)}^2)$

I am trying to evaluate the sum here , $$\sum_{i=1}^n \sum_{j=i+1}^n({(x_i-x_j)}^2+{(y_i-y_j)}^2)$$ How do this sum can be simplified to $$n\sum_{i=1}^n({x_i}^2+{y_i}^2) - {(\sum_{i=1}^n{x_i})}^2-{(\sum_{i=1}^n{y_i})}^2$$ . I need to understand…
Tamim Addari
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Average length of a bitstring

I am trying to compute the average length of a bit string from all bit strings of $\{0,1\}^n$. By length n I mean a bit string of length n where the most significant bit is 1. I know there is $2^0$ strings of length 1, $2^1$ strings of length 2 ...…
digy
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Summation of indicator function

I need to calculate this summation. I have tried to solve it myself but can't seem to get anywhere. I know that the answer needs to be $2q+1-h$. $$\sum_{j, k=-q}^q 1_{(h+j-k=0)}$$
user169475
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Question related to the sigma during expectation in probability

How is this below possible: $ \sum_{i=1}^\infty 1/i = \infty $
Abhimanyu Aryan
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Symmetric Sum Notation

From here is the following excerpt Suppose one is given a homogeneous symmetric polynomial $P$ and asked to prove that $P(x_1, \ldots , x_n) ≥ 0$ How should one proceed? Our first step is purely formal, but may be psychologically helpful. We…
RE60K
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Question about partial sum

I'm confused on the partial sums formula Why is $$\sum_{i=m+1}^\infty \frac{2}{3^i}=\frac{1}{3^m},$$ if $$\sum\limits_{k = 0}^\infty\frac{2}{3^k} = 3.$$
user164179
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How to simplify an expression where the summation is over all subsets of a given set?

I want to simplify this expression: $\sum\limits_{\emptyset \ne I \subseteq \mathbb{N}_k}(-1)^{|I|-1}|A_I|+|A_{k+1}|-\sum\limits_{\emptyset \ne I \subseteq \mathbb{N}_k}(-1)^{|I|-1}|A_I \cap A_{k+1}|$ To this expression: $\sum\limits_{\emptyset \ne…
mauna
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Easy question regarding this proof

I do not understand a small step in a proof I'm reading at the moment. Why are the following things equal? $$\sum_{k=1}^{n} \frac{1}{2k-1} - \frac{1}{2} \sum_{k=1}^{n} \frac{1}{k} = \sum_{k=1}^{2n} \frac{1}{k} - \sum_{k=1}^{n} \frac{1}{k}$$
rehband
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Why is $\sum_{i=0}^{n}1=(n+1)$?

Why is $$\sum_{i=0}^{n}1=(n+1)?$$ I mean, as 1 does not depend on $i$, so shouldn't be the sum equals to 1 (as I was adding nothing, just keeping 1)?
mvfs314
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How do we approach this summation question?

How do we find the following sum--- $\displaystyle\sum_{m=1}^\infty \displaystyle\sum_{n=1}^\infty \frac{mn^2}{2^n(n2^m+m2^n)}$
user1001001
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How do I find the partial sum of this

So I have a sum defined below: $$ \sum_{m=1}^n 2^{-m} $$ I know the partial sum equals $$ \frac{1}{2^n}(2^n - 1)\ $$ But how do you go from one to the other?
user160114
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Evaluate $S=\Sigma\Sigma\Sigma x_ix_jx_k$

I am a novice in this type of sums and I can't even understand the meaning of the three sigmas. Somehow, I am guessing that the answer might be $0$ but I am not sure. I need a well-explained answer(with examples) explaining the meaning and the…
User Not Found
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Samplification of a sum of multiplication

Supposing I have the following sequence based on two indexes: $a$ and $b$. For $a$ starting with $1$ and $b$ starting with $5$ we have the following sum: $$1 \cdot 5 + 2 \cdot 4 + 3 \cdot 3 + 4 \cdot 2 + 5 \cdot 1$$ i.e. $a$ is incremented from…
Arteo
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Multiplying infinite sums

Is that true for infinite sums that $c\cdot \sum_{n=1}^{\infty} a_n=\sum_{n=1}^{\infty} c\cdot a_n$?Or it only applies when the sum is finite($\sum a_n \lt \infty$)?
user128576
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