All Questions
19
questions
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Closed form for $ \sum_{a_1=0}^\infty~\sum_{a_2=0}^\infty~\cdots~\sum_{a_n=0}^\infty \dfrac1{(a_1!+a_2!+\ldots+a_n!)} $ [closed]
After reading this post and the general solution for that case, I wonder if there is a closed form for the general solution for this sum:
$ \sum_{a_1=0}^\infty~\sum_{a_2=0}^\infty~\cdots~\sum_{a_n=0}^\...
- 923
0
votes
2
answers
125
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Pi/product notation property applications problem
I have recently attempted to simplify this
$$ P(n) = \prod_{v=2}^{n} (2 + \frac{2}{v^2 - 1}) $$
I have reached an answer (which is wrong) through the following steps:
rearranging what is inside the ...
- 15
1
vote
0
answers
46
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Is this summation equality true
A double factorial is such that $(2n)!!=(2n)(2n-2)...(2),$ and $(2n-1)!!=(2n-1)(2n-3)...3$. So $8!!=(8)(6)(4)(2)$. More information here: https://en.wikipedia.org/wiki/Double_factorial
With this in ...
- 79
1
vote
1
answer
69
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Determine if the series representation is true or not
A double factorial is such that $(2n)!!=(2n)(2n-2)...(2),$ and $(2n-1)!!=(2n-1)(2n-3)...3$. So $8!!=(8)(6)(4)(2)$. More information here: https://en.wikipedia.org/wiki/Double_factorial
With this in ...
- 79
1
vote
1
answer
34
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Question about the Limit of a Sum of a Series
I am a Calculus II student and we have been learning about series, summation, factorials, etc. I thought about this while I was in the shower and I cannot seem to get it out of my head.
What would ...
- 45
2
votes
1
answer
132
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On the generalized Leibniz rule
problem definition
I have to evaluate in $z=0$ the $n$-th derivative with respect $z$ of the product $f(z)\cdot z^k$, where $f(\cdot)$ is a generic smooth function and $k$ is a given integer. I will ...
- 479
3
votes
2
answers
518
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Differentiation involving sigma notation
I am having trouble understanding the following relationship in one of my assigned problems:
$$\dfrac{d}{dx}\sum_{n = 0}^\infty \dfrac{x^n}{(n + 1)!} = \sum_{n = 1}^\infty \dfrac{nx^{n-1}}{(n + 1)!}$$
...
- 133
2
votes
0
answers
98
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How to calculate this binomial sum?
In a previous post i was asking about a complicated sum which seems not possible to simplify:
Is it possible to simplify this sum?
I want now to calculate the simpler sum:
$$ S := \sum_{k=1}^{n} \...
- 31
1
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0
answers
103
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Is it possible to simplify this sum?
I need help to simplify this sum:
$$ \sum_{k=1}^{n} \binom{n}{k} ((k-1)!)^2 \left[ \binom{n}{k} - \binom{n-k}{k} \right]$$
I have tried to use Pascal's formula for the difference:
$$\binom{n}{k} - \...
- 31
2
votes
2
answers
45
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Sum expressed as a function
Is there any way to express $\sum_{n=0}^{x-2} \frac {1}{x-n} $ as a function of $x$, as opposed to being a summation? I tried doing this through modelling on Desmos, but the closest I could get was $y=...
- 21
1
vote
1
answer
57
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Computation of an integral involving summation over factorials.
For the computation of an expected value, with $m>i$ and $m,i\in\mathbb{N}$, I am trying to show that
$$
\int_0^1 x\cdot m\binom{m-1}{i-1}x^{i-1}(1-x)^{m-i}\mathrm{d}x=\frac{i}{m+1}.
$$
However, ...
- 819
3
votes
1
answer
75
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How can I prove that: $\forall n\geq 2 $ $ \sum_\limits{k=2}^{n}\frac{1}{\log_{k}{n!}}=1$
$\forall n\geq 2 $ $ \sum_\limits{k=2}^{n}\frac{1}{\log_{k}{n!}}=1$
I worked it out for $n=3$:
$\sum_\limits{k=2}^{n}\frac{1}{\log_{k}(n!)}=\frac{1}{\log_{2}(3)+\log_{2}(2)}+\frac{1}{\log_{3}(3)+\...
- 1,559
0
votes
2
answers
68
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Integral To Summation Problem
$\int x^n e^{cx}\; \mathrm{d}x = \frac{1}{c} x^n e^{cx} - \frac{n}{c}\int x^{n-1} e^{cx} \mathrm{d}x = \left( \frac{\partial}{\partial c} \right)^n \frac{e^{cx}}{c} = e^{cx}\sum_{i=0}^n (-1)^i\,\frac{...
- 11
1
vote
1
answer
363
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Proof By Induction With Integration Problem
I am required to prove this formula by induction$$ \int x^k e^{\lambda x} = \frac{(-1)^{k+1}k!}{\lambda^{k+1}} + \sum_{i=0}^k \frac{(-1)^i k^\underline{i}}{\lambda^{i+1}}x^{k-i}e^{\lambda x}$$
where $...
- 11
2
votes
3
answers
114
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Is $n! \sum_{i=0}^n{\frac{(-1)^i}{i!}}- (n-1)! \bigg[\sum_{i=0}^{n-2}{\frac{(-1)^i}{i!}}+...+\sum_{i=0}^{2}{\frac{(-1)^i}{i!}}\bigg]=(n-1)!$ true?
I am in the middle of doing a problem and has this sort of expression. I have a feeling that the following equality holds:
$$n! \sum_{i=0}^n{\frac{(-1)^i}{i!}}- (n-1)! \bigg[\sum_{i=0}^{n-2}{\frac{(-1)...
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