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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Calculate sum of a row $\sum_{k=1}^{n}\left (2k-1\right )/2^k$

Please help to solve: $\sum_{k=1}^{n}\left (2k-1\right )/2^k$ I tried to separate the numerator into $2k$ and $-1$, and the second fraction is easy to calculate. But I can't get sum for the first one. I know the answer $3-(2n+3)/2^n$ , but can't get…
DDR
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Simplification. Difficulty in detecting the series $\sum_{i=0}^\infty \frac{x^i}{i!} = e^x$

Any suggestion on how to simplify the following? $$\sum_{m=k+1}^\infty \frac{((1-p)\lambda)^m}{(m-k)!}$$
Strictly_increasing
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Easy double sum trouble : $\sum_{1 \leq i,j \leq n} (i+j)^2$

There's a mistake somewhere but I really can't see where. $\sum_{1 \leq i,j \leq n} (i+j)^2 = \sum_{i=1}^n \sum_{j=1}^n \left( i^2+2ij+j^2\right) = \sum_{i=1}^n \left( \sum_{j=1}^n i^2 + 2i\sum_{j=1}^n j + \sum_{j=1}^n j^2\right) $ $=…
Tanguy Freycon
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Finding the sum of $rx^{r-1}$

May I know how to find $rx^{r-1}$. I searched for it, it said that we can utilize $\frac{d}{dx}(1-x)^{-1} =\frac{d}{dx} (1-x)^{-2}=\frac{d}{dx} \sum {x^t} =\sum {tx^{t-1}} $ Why is this true? Thank you very much for your reply.
Henry Cai
  • 633
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Discrete Summation transformation

Tried to prove this by writing out as I hoped the e terms would simplify, but it wasn't the case so I am stuck now. Am I missing a summation property I could use? $$\sum_{n=0}^{n=5}(\frac{1}{2}e^{\frac{-i2\pi k}{6}})^n…
dsisko
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Why can a double sum be rearranged like this?

$$\sum_{r = 1}^n \sum_{p=1}^n A_{j,p}A_{p,r}A_{r,k}=\sum_{p = 1}^n \sum_{r=1}^n A_{j,p}A_{p,r}A_{r,k}$$ Would reviewing double sums help me to avoid asking questions like this? Does anybody know of a good source I can use to learn properties of…
user736276
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Why $\sum_{x=y}^n 1 = n-y+1 ?$

Why $\sum_{x=y}^n 1 = n-y+1 ?$ We know that $1 \leq y \leq x \leq n$. $$\sum_{x=y}^n 1$$ Is there a formula for solving this? I would have said that it should equal n-y, but why the +1?
Meghan A.
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Proof that $\sum(\frac{(-1)^n}{\sqrt{n}} + \frac1n)$ diverges

I have $\sum(\frac{(-1)^n}{\sqrt{n}} + \frac1n)$. Nth test: $\lim_{n->\infty}{(\frac{(-1)^n}{\sqrt{n}} + \frac1n)}$ = 0. I think that we can not split it into two sums like $\sum(\frac{(-1)^n}{\sqrt{n}}) + \sum(\frac1n)$ because the second one is…
IlFiltsin
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prove: $\sum_{r=0}^{ ∞}\frac{{{r+k}\choose{k}}}{(r+k)(r+k-1)}x^r=\frac{\left(k- 2\right)!}{k!}\cdot\frac{1}{\left(1-x\right)^{\left(k-1\right)}}$

prove: $$\sum_{r=0}^{ ∞}\frac{{{r+k}\choose{k}}}{(r+k)(r+k-1)}x^r=\frac{\left(k- 2\right)!}{k!}\cdot\frac{1}{\left(1-x\right)^{\left(k-1\right)}}$$ I tried to expand the sum such that: $$\sum_{r=0}^{…
Absurd
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Trying to understand index shifting

I don't understand index shifting/general rules for Summations. $S_n := \frac{1}{1+n}+\frac{1}{1+(n+1)}+...+\frac{1}{2n}$. Finding a Summation formula for this equals either: $A(n):=\sum_{k=n}^{2n-1} \frac{1}{1+k}$ or $S(n):=\sum_{k=n+1}^{2n}…
Lisa
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How to find the sum $\sum_{k = 1}^n \frac{1}{(k+1)\sqrt{k}+k\sqrt{(k+1)}}$?

How do you find the sum $$\sum_{k = 1}^n \frac{1}{(k+1)\sqrt{k}+k\sqrt{(k+1)}}?$$ Sorry for not provide any of my idea since I have no idea about what to do. I hope everyone here could help me. Thank you
user635988
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Are these two summations equal?

Is the left hand side equal to the right hand side here? If so, why? $$\sum_{i=1}^n \frac{1}{i} + 2\cdot\sum_{1\le i\lt j\le n}\frac{1}{ij} = \left(\sum_{i=1}^n \frac{1}{i}\right)^{\!2}+\sum_{i=1}^n \frac{1}{i} - \sum_{i=1}^n \frac{1}{i^2}$$ Thanks!
Jdo
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comparing two equations and obtaining some results

Assuming the following equation $$ \sum_{m=0}^M b_m r^m \left[\sum_{i,j}i j\lambda_i \lambda_j r^{i+j-2}-\sum_j j(j+1) \lambda_j r^{j-2}+2E\right]-2\sum_{k=0}^K a_k r^k=0 \tag{1} $$ the authors in a paper says that Equating the coefficients of…
Wisdom
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Differentiating summations

Consider the following equation $$ \sum_{m=0}^{M}b_m r^m \left(\frac {d^2}{dr^2}+\frac{2}{r}\frac d {dr}+2E\right)\Psi-2\sum_{k=0}^{K}a_k r^k \Psi=0 \tag{1} $$ Now suppose $$ \Psi(r)=A \exp{[-S(r)]} \tag{2} $$ where $$ S(r)=\sum_{n=1}^N \lambda_n…
Wisdom
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How to determine the number of terms of a summation for a given expression

Suppose I have a potential as form $$ V(r)=\frac{\sum_{k=0}^{K}a_k r^k}{\sum_{m=0}^{M}b_m r^m} \tag{1} $$ How can I match the given potential $$ V(r)=\frac{-Z}{r}+\alpha r+\beta r^2 \tag{2} $$ with equation (1)? I mean how can I find the values of…
Wisdom
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