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7
questions with no upvoted or accepted answers
2
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answers
67
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Closed form for $\psi^{1/k}(1)$, where $k$ is an integer
I have proven the identity
$$
\sum_{k=1}^{\infty} \dfrac{\operatorname{_2F_1}(1, 2, 2-1/t,-1/k)}{{k}^{2}} = Γ(2-\dfrac{1}t){\psi^{1/t}(1)}+\psi(-\dfrac{1}t)(\dfrac{1}t(1-\dfrac{1}t))+\gamma(1-\dfrac{1}...
2
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0
answers
82
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Limit : $\lim_{n\to+\infty}a^n(n-\zeta(2)-\zeta(3)-\cdots-\zeta(n))$
question
Compute the limit $$\displaystyle{\lim_{n\to+\infty}a^n(n-\zeta(2)-\zeta(3)-\cdots-\zeta(n))}$$, if any, for the various values of the positive real a, where $\zeta$ the zeta function of Mr. ...
2
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92
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How to find $\sum_{n \in \mathbb Z_+} \frac{2^{n-1}}{2^{2^n}}$?
I'm trying to calculte the measure of a fat Cantor set, but run into this question:
How to find $$\sum_{n \in \mathbb Z_+} \frac{2^{n-1}}{2^{2^n}}$$
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1
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116
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Sum to closed form?
I need to evaluate the following summation to closed form:
$$\sum _{k=1}^{\infty } \frac{\sin (e k n \pi ) \sin (e k (1+n) \pi )}{k \pi \sin (e k \pi )}$$
where: $n>1$ and $n\in \mathbb{Z}$, $e \...
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1
vote
0
answers
39
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Closed form for $\sum_{k\in\mathbb{N}}\frac{k}{a\uparrow^kb}$
Let $a,b\in\Bbb{N}$. Is there a closed form for $\displaystyle\sum_{k\in\mathbb{N}}\frac{k}{a\uparrow^kb}$ ? (I use Knuth's up arrow notation)
If so, how can we obtain it ?
If there isn't a closed ...
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0
votes
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132
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Calculation of $\sum_{n=1}^{\infty}\frac{\psi_1(n)}{2^nn^2}$
Calculation of $$\sum_{n=1}^{\infty}\frac{\psi_1(n)}{2^nn^2}$$
My attempt
\begin{align*}
\sum_{n=1}^\infty\frac{\psi_1(n)}{2^n n^2} &= -\sum_{n=1}^\infty\psi_1(n)\left(\frac{\log(2)}{2^n n}+\int_0^...
0
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173
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Manipulation of summations
this question branches off another question that can be seen here
Now we begin be taking a look at the following expressions:
$$
\sum_{k=1}^{n-l} \sum_{j-0}^m \frac{\ln(g)^{m-j}}{g^k} \frac{d^j}{df^j}...
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