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 find this sum $f(x)=\sum_{n=1}^{\infty}\frac{x^{n^2}}{n}$

First, Merry Christmas everyone! Find this sum $$f(x)=\sum_{n=1}^{\infty}\dfrac{x^{n^2}}{n},1>x\ge 0 \tag{1}$$ This problem is creat by Laurentiu Modan.and I can't see this solution. I know this…
math110
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A finite summation involving $2013$

Can you help me compute the summation below? $$1+\frac{1}{2}+\cdots+\frac{1}{2013}+\frac{1}{1\cdot2}+\frac{1}{1\cdot3}+\cdots+\frac{1}{2012\cdot 2013}+\cdots+\frac{1}{1\cdot2\cdots2013}$$
Saul of Tarsus
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Approximating $\sum_{k=1}^{\infty}\frac{\sin(k^{1/n})}{k}\text{ for }n\in\mathbb{N}$

Again, inspired by this question, and the great answers I received here, I am curious as to why these infinite sums can be modelled with smooth functions. It appears that $\sum_{k=1}^{n}\dfrac{\sin(k)}{k}$ can be modelled with…
martin
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How find this interesting sum $\sum_{n=1}^\infty\ln(f(n))+\ln(g(n))$

How find this following interesting sum $$\sum\limits_{n = 1}^\infty {\left\{ {{{\ln }^2}\left( {n - \frac{1}{4}} \right) + {{\ln }^2}\left( {n{\rm{ + }}\frac{1}{4}} \right) - {{\ln }^2}\left( {n - \frac{1}{2}} \right) - {{\ln }^2}\left( {n +…
math110
  • 93,304
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Is it possible to swap sums like that?

Say that I have two sums like this : $$\sum_{a=0}^n\sum_{b=0}^m f_{ab}$$ Would it be true to say that this expression can be considered as equal to : $$\sum_{a=0}^m\sum_{b=0}^n f_{ab}$$ As long as the expression that comes after the sums is the same…
Pop Flamingo
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How to prove that $ \sum_{k=1}^{n-1} \frac{1}{1-e^{2 \pi i k/n}} = \frac{n-1}{2}$?

I came across the fact that $$ \sum_{k=1}^{n-1} \frac{1}{1-e^{2 \pi i k/n}} = \frac{n-1}{2}.$$ How can we prove this identity?
user111187
  • 5,856
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If the earth's rotational speed increased by 2% each day starting today…what would be the difference in age 20 years from now?

If the new adjusted revolution of the earth still equaled one day and 365 days still equaled one year, how old would someone be 20 years from now (20 years based on the current rotation of the earth) compared to the new rotation of the earth? I'm…
SUM GUY
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Donald Knuth and algebraic operations on sums

I'm reading through Knuth's first book in the TAOCP series and I'm on chapter 1.2.3 (Sums and Products). As this is my first encounter with summation I did take the time to go through what I could find on Khan Academy. I've gone through the chapter…
Ixmatus
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How do you "simplify" the sigma sign when it is raised to a power?

How do you simplify the following expression: $$\left(\sum^{n}_{k=1}k \right)^2$$ I am supposed to show that $$\left(\sum^{n}_{k=1}k \right)^2 = \sum^{n}_{k=1}k^{3} $$ The problem is I do not really know how to manipulate the sigma sign. I know…
user376984
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Summation of reciprocals

$$\sum_{i=1}^n \frac1{n+i} = \sum_{i=1}^{2n}\frac{(-1)^{i+1}}i,\quad \text{ for }n\ge 1.$$ I really have no clue how on to solve this one. It's the last one on my assignment. Would really enjoy some help with this one.
Ajax Edm
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"Commutativity" of sums

In general, is $\sum_i \sum_j f(i,j) = \sum_j\sum_i f(i,j)$ ? With $f(i,j)$ I mean some expression that depends on $i$ and $j$. If yes, how could I prove that?
Christian
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How to show that $\sum_{k=1}^n k(n+1-k)=\binom{n+2}3$?

While thinking about another question I found out that this equality might be useful there: $$n\cdot 1 + (n-1)\cdot 2 + \dots + 2\cdot (n-1) + 1\cdot n = \frac{n(n+1)(n+2)}6$$ To rewrite it in a more compact way: $$\sum_{k=1}^n…
Martin Sleziak
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Calculate the sum : $S=1+\frac{\sin x}{\sin x}+\frac{\sin2x}{\sin^2 x}+\cdots +\frac{\sin nx}{\sin^n x}$

I was given a task to calculate this sum: $$S=1+\frac{\sin x}{\sin x}+\frac{\sin 2x}{\sin^2 x}+\cdots +\frac{\sin nx}{\sin^n x}$$ but I'm not really sure how to start solving it. Like always, I would be grateful if someone could provide a subtle…
Transcendental
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How find this $\sum_{n=1}^{\infty}\frac{(-1)^{n-1}\zeta_{n}(3)}{n}=?$

Question: show that $$\sum_{n=1}^{\infty}\dfrac{(-1)^{n-1}\zeta_{n}(3)}{n}=\dfrac{19\pi^4}{1440}-\dfrac{3}{4}\zeta{(3)}\ln{2}?$$ where $$\zeta_{n}(3)=\sum_{k=1}^{n}\dfrac{1}{k^3}$$ But I use this computer find this and my reslut is wrong? Thank…
math110
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How to derive the formula for the sum of the first $n$ odd numbers: $n^2=\sum_{k=1}^n(2k-1).$

How to derive this formula? $$n^2=\sum_{k=1}^n(2k-1).$$
user115735
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