All Questions
46
questions
1
vote
2
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9k
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Find the closed form of a summation from $k=1$ to $n$
For my discrete mathematics class, I need to express this summation in closed form in terms of $n$,
$$\sum_{k=1}^n \left(6 + 2 \cdot \frac{k}{n}\right)^2. $$
I was in the hospital when they went ...
- 363
14
votes
3
answers
2k
views
Closed form for $\sum_{n=1}^{\infty}\frac{1}{\sinh^2\!\pi n}$ conjectured
By trial and error I have found numerically
$$\sum_{n=1}^{\infty}\frac{1}{\sinh^2\!\pi n}=\frac{1}{6}-\frac{1}{2\pi}$$
How can this result be derived analytically?
- 2,903
3
votes
3
answers
3k
views
Sum of a series $\frac {1}{n^2 - m^2}$ m and n odd, $m \ne n$
I was working on a physics problem, where I encountered the following summation problem:
$$ \sum_{m = 1}^\infty \frac{1}{n^2 - m^2}$$ where m doesn't equal n, and both are odd. n is a fixed constant
...
- 539
2
votes
3
answers
92
views
Prove that for any positive integer $n$ and $d$, $\sum_{k=0}^d 2^k\log_2(\frac{n}{2^k})=2^{d+1}\log_2(\frac{n}{2^{d-1}})-2-\log_2{n}$
I could prove it by induction, but I need to see how I might have discovered it for myself (cause that's what's gonna be on exam).
- 813
1
vote
0
answers
39
views
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 ...
- 7,770
2
votes
0
answers
92
views
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 $$\sum_{n \in \mathbb Z_+} \frac{2^{n-1}}{2^{2^n}}$$
- 407
0
votes
1
answer
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:
$$
\sum_{k=1}^{n-l} \sum_{j-0}^m \frac{\ln(g)^{m-j}}{g^k} \frac{d^j}{df^j}...
- 1,957
14
votes
2
answers
482
views
Closed-form of $\sum_{n=0}^\infty\;(-1)^n \frac{\left(2-\sqrt{3}\right)^{2n+1}}{(2n+1)^2\quad}$
The following question is purely my curiosity. During my calculation to answer @Chris'ssis's question in chat room I encountered this series
$$\sum_{n=0}^\infty\; \frac{\left(2-\sqrt{3}\right)^{2n+1}}{...
- 11k
0
votes
1
answer
2k
views
How to solve integrals using series?
Many places I have seen when solving integrals you change a lot of it into sums.
Finding $\int_{0}^{\pi/2} \frac{\tan x}{1+m^2\tan^2{x}} \mathrm{d}x$
Is just an example.
So in general, how do you ...
- 11.2k
6
votes
2
answers
288
views
Closed-form of $\sum_{k=0}^{\infty} \frac{k^a\,b^k}{k!}$
While working on this question I think I've found a closed-form expression for the following series, but I don't know how to prove it.
Let $a \in \mathbb{N}$ and $b \in \mathbb{R}$. Then
$$\sum_{k=0}...
- 12.4k
6
votes
2
answers
242
views
How to prove $\sum_{n=0}^{\infty} \frac {(2n+1)!} {2^{3n} \; (n!)^2} = 2\sqrt{2} \;$?
I found out that the sum
$$\sum_{n=0}^{\infty} \frac {(2n+1)!} {2^{3n} \; (n!)^2}$$
converges to $2\sqrt{2}$.
But right now I don't have enough time to figure out how to solve this.
I would ...
- 63
3
votes
5
answers
390
views
Infinite Series $\left(\frac12+\frac14-\frac23\right)+\left(\frac15+\frac17-\frac26\right)+\left(\frac18+\frac{1}{10}-\frac29\right)+\cdots$ [duplicate]
How do I find the sum of the following infinite series:
$$\left(\frac12+\frac14-\frac23\right)+\left(\frac15+\frac17-\frac26\right)+\left(\frac18+\frac{1}{10}-\frac29\right)+\cdots$$
The series ...
- 375
0
votes
1
answer
189
views
How to solve this summation (Lerch Transcendent)?
How is it possible to deduce the closed form of the following?
$$\sum_{i = 0}^{n - 1} \frac{2^i}{n - i} = ?$$
6
votes
5
answers
512
views
Infinite Series $1+\frac12-\frac23+\frac14+\frac15-\frac26+\cdots$
Was given the following infinite sum in class as a question, while we were talking about Taylor series expansions of $\ln(1+x)$ and $\arctan(x)$:
$$1+\frac12-\frac23+\frac14+\frac15-\frac26+\cdots$$
...
- 61
11
votes
2
answers
689
views
Infinite Series $\sum\limits_{k=1}^{\infty}\frac{k^n}{k!}$
How can I find the value of the sum $\sum_{k=1}^{\infty}\frac{k^n}{k!}$?
for example for $n=6$, we have
$$\sum_{k=1}^{\infty}\frac{k^6}{k!}=203e.$$
- 5,626