Questions tagged [summation]
Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.
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$\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$ as a limit of a sum
Working on the same lines as
This/This and
This
I got the following expression for the Dilogarithm $\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$:
$$\operatorname{Li}_{2} \left(\frac{1}{e^{\...
- 896
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54
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$Li_{2} \left(\frac12 \right)$ vs $Li_{2} \left(-\frac12 \right)$ : some long summation expressions
Throughout this post, $Li_{2}(x)$ refers to Dilogarithm.
While playing with some Fourier Transforms, I came up with the following expressions:
$$2 Li_{2}\left(\frac12 \right) + \frac{\pi^{2}}{6} = \...
- 896
2
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2
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162
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Is there a closed formula for this sum? [duplicate]
The sum is $$f_{n}=\sum_{k=1}^{n}\arctan\left(\frac{1}{\sqrt{k}}\right)$$
I figured I need a closed formula for this or for the cosine of this whole expression in order to get a polar representation ...
- 155
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33
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Change order of summation
I need to change the summation order in the sum:
$$
\sum\limits_{n=0}^{\infty} \sum\limits_{k=0}^{ n m } F(k, n - mk)
$$
In Srivastava's book, I came across a similar formula
\begin{array}{c}
\sum\...
- 8,194
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2
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54
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Upper bound on the finite sum of $\sum_j x^j/j!$ [closed]
How can I derive an upper bound on the following finite summation,
\begin{equation}
S = \sum_{j=1}^k \frac{x^j}{j!},
\end{equation} where $0 < x$, in terms of $x$ and $k$ (it's perfectly fine to ...
- 23
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63
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Showing that $\sum_{i = 1}^N \sum_{j = N - i + 1}^N \frac{1}{ij} = \sum_{i = 1}^N \frac{1}{i^2}$ without induction
It's straightforward to show via induction that
$$
\sum_{i = 1}^N \sum_{j = N - i + 1}^N \frac{1}{ij} = \sum_{i = 1}^N \frac{1}{i^2}\text{.}
$$
(For example, given the table
$$
\begin{array}{|c|c|c|c|}...
- 771
-2
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56
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Asymptotic notation equation problem [closed]
Hey guys this is question that we should say if both sides are equal or not and i want yall to help me find my mistake
I mentioned my question
Where is my answer’s problem?
my answer
- 15
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3
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194
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Summation of arithmetic series
How to represent a sum of $n$ items of an arithmetic series with the use of the sigma notation?
For the sum of the 10 first items, is the below one correct?
$$
\color{gray}{
\sum_{\{{x_i: x_{i-1}+k \}}...
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2
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Reference for ${}_nC^0 + {}_nC^1 + {}_nC^2 + \cdots + {}_nC^n = 2^n$ [closed]
How can I find, or what is a good reference for:
$${}_nC^0 + {}_nC^1 + {}_nC^2 + \cdots + {}_nC^n = 2^n$$
I could write
References
[1] Binomial sums, Binomial Sums -- from Wolfram MathWorld
but I need ...
- 15
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0
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51
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Derivative with respect to an index within a summation
First note that
$$ \sum_{n=0}^{\infty} \frac{\left(\frac{1}{2}\right)_{n}}{n!} \, x^n = \frac{1}{\sqrt{1-x}}, $$
where $(a)_{n}$ is the Pochhammer notation. The background is how to evaluate the ...
- 26.5k
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1
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58
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What is $\sum_{m=0}^{\lfloor k/3\rfloor}{2k\choose k-3m}$ [closed]
With the floor function, I am not sure how to approach this.
Edit: I have a formula $\frac{1}{6}\left(4^k+2+3\frac{(2k)!}{(k!)^2} \right)$, but I got it purely from guess work.
- 101
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Closed form of summation involving square root [closed]
Does anyone have any idea how to find a closed form for the following summation? I cannot for the life of me figure anything out.
$$\sum_{i=1}^n(n-i)\sqrt{i^2+1}$$
7
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1
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364
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Which numbers are sums of finite numbers of reciprocal squares?
Question: Is there a “nice” characterization of rational numbers $q$ for which $q$ can be written as
$$q = \frac{1}{n_1^2} + \frac{1}{n_2^2} + \dots + \frac{1}{n_k^2}$$
for distinct natural numbers $...
- 10.5k
-1
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1
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Resources to master summation symbol [closed]
I noticed that I have some difficulties to use the summation tools( change of index, double or multiples summation...). Do you have some resources or book to master this topic. I am using concrete ...
- 9
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54
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Closed form solution for $\displaystyle \sum _{i=r+1}^{k}\frac{1}{i-1} \cdot \frac{( n-i) !}{( k-i) !}$, where $n,k,r$ are constants and $r \leq k<n$ [closed]
Context
I arrived at this summation, while computing a probability of The Secretary Problem.
Question:
Find Closed form for this or if possible simplify
$$
\frac{r}{n}\sum_{i = r + 1}^{k}\frac{1}{i - ...
- 17