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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Sum $\sum_{x=1}^n\sum _{y=1}^{x-1}\frac{1/2^x*1/2^y}{1/2^x+1/2^y}$

Is there a way to calculate following summation $\sum_{x=1}^n\sum _{y=1}^{x-1}\frac{1/2^x*1/2^y}{1/2^x+1/2^y}$ Can it be reduced to something simple?
user3639559
  • 61
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Efficient Way of finding possible sums in equation

I am looking for an efficient way to find the following numbers numbers $a, b$ Let $x$ be 63 (as example) $$\sum_{k = a}^b k = \sum_{l = 1}^b l - \sum_{j=1}^{a-1} j = \frac{b*(b+1)}{2}-\frac{a*(a-1)}{2} = x = 63$$ in this case for example $$63 =…
Entimon
  • 141
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Find the exact closed from expression of $1^2 + 3^2 + 5^2 + · · · + (2n + 1)^ 2$

I know the above expression equals to $\frac{n(2n−1)(2n+1)}{3}$, but how exactly can i come up with something from scratch?
Benjamin
  • 123
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How to find $\sum_{k=1}^n k^k$?

Actually question which I found: Find the sum of the series $1^1+2^2+3^3+ \cdots +n^n $ This question has been bothering me since a long time. Any help would be appreciated!
Harshal Gajjar
  • 1,664
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Equivalence of summations

Show that $$\frac{1}{n}\sum^{n}_{i=1} (x_{i} - \bar{x})^{2}\equiv \frac{1}{n}\sum^{n}_{i=1}x_{i}^{2} - \bar{x}^{2}.$$ Note that $\bar{x} = \frac{1}{n}\sum^{n}_{i=1} x_{i}$. So I have started by: \begin{align} \frac{1}{n}\sum^{n}_{i=1} (x_{i} -…
user2850514
  • 3,689
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Evaluate the sum.

Evalute the following sum: $ 1 + 2 + 2 + 3 + 4 + 4 + 5 + 6 + 6 + . . . .+ (n-1) + n + n$. I tried doing it but I keep getting the wrong answer. I've used known sums to solve it.
George J. Adams
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Finding a number $M$ such that $S_N<10^{-20}$ for all $N>M$

Given that $$u_n=\frac{1}{n^2-n+1} - \frac{1}{n^2+n+1}$$ $$S_n=\sum_{n=N+1}^{2N} u_n$$ Find a number $M$ such that $S_N<10^{-20}$ for all $N>M$ I did: $$S_N = \sum_{n=N+1}^{2N} u_n = \sum_{n=0}^{2N} u_n - \sum_{n=0}^{N} u_n$$ Using method of…
The Artist
  • 3,064
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Search for summation formula

Is there any closed formula for the sum $ ~\sum_{k = 1} ^ {n} r^k k^r ~$ where $~r~$ is an integer? Thank you very much in advance.
Igor
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Nice derivation of $\sum_{n=1}^\infty \frac{1}{n} \left( \frac{q^{2n}}{1-q^n}+\frac{\bar q^{2n}}{1-\bar q^n}\right)=-\sum_{m=2}^\infty \ln |1-q^m|^2$

I'm searching for a nice derivation of the formua $\sum_{n=1}^\infty \frac{1}{n} \left( \frac{q^{2n}}{1-q^n}+\frac{\bar q^{2n}}{1-\bar q^n}\right)=-\sum_{m=2}^\infty \ln |1-q^m|^2$ given for example in http://arxiv.org/abs/arXiv:0804.1773 eq.(4.27).
ungerade
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How to compute $\sum_{k =1}^{100}(-1)^k$

Today I tried to compute $$ \sum_{k =1}^{100}(-1)^k $$ Is there a way to find the result more quickly ? Below if my attempt to find the result. Especially without considering the case of odd and even numbers like I did ? Let's consider the…
alexandrekow
  • 247
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What is the sum of $\sum_{n=1}^\infty (n^2+n^3)x^{n-1}$?

Consider the power sequence $$\sum_{n=1}^\infty (n^2+n^3)x^{n-1}$$ What is the function to which it sums to? My reasoning is to differntiate the sum with respect to $x$, then to integrate with respect to x from $0$ to $x$ after variation of the sum…
E Be
  • 550
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Summation of series

Find $\sum_1^n$ $\frac {2r+1}{r^2(r+1)^2}$ Also, find the sum to infinity of the series. I tried decomposing it into partial fractions of the form $\frac Ar$ + $\frac{B}{r^2}$ + $\frac{C}{(r+1)}$ + $\frac{D}{(r+1)^2}$ but it was getting too…
user140161
  • 1,363
2
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5 answers

How to describe 4-8+12-16+20-24+28 using summation ($\Sigma$) notation?

Can anyone find the ∑ summation for this please? 4-8+12-16+20-24+28 It seems to be going up by steps of 4, but I can't seem to get how I should write it down, since it uses both + and -
Tim
  • 23
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4 answers

Sum of a series problem involving cubes

For any odd integer $n$, evaluate $n^3 - (n-1)^3 + \cdots + (-1)^{n-1} \cdot 1^3 = \sum_{k=1}^n (-1)^{k+1}\cdot k^3$ How would you go about solving such a problem? Any help would be appreciated.
user34304
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3 answers

Find the sum of this series: $\sum_{n=1}^\infty$ $\dfrac{n}{2^{n-1}}$

May I know how I should go about finding the sum of this series? $\displaystyle\sum_{n=1}^\infty$ $\dfrac{n}{2^{n-1}}$ I am really stuck. Thanks!
thbcm
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