Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.
20
votes
3answers
398 views
Finding the value of $\sum\limits_{k=0}^{\infty}\frac{2^{k}}{2^{2^{k}}+1}$
Does this weighted sum of reciprocals of Fermat numbers,
$$
F=\sum_{k=0}^{\infty}\dfrac{2^{k}}{2^{2^{k}}+1}
$$
have a nice closed form? Wolfram says it's $1$.
Thanks.
17
votes
6answers
896 views
Is there a formula for $\sum_{n=1}^{k} \frac1{n^3}$?
I am searching for the value of $$\sum_{n=k+1}^{\infty} \frac1{n^3} \stackrel{?}{=} \sum_{n = 1}^{\infty} \frac1{n^3} - \sum_{n=1}^{k} \frac1{n^3} = \zeta(3) - \sum_{n=1}^{k} \frac1{n^3}$$
For which ...
16
votes
3answers
276 views
Counting binary sequences with no more than $2$ equal consecutive numbers
I invented the following problem, but I can't solve it.
There are $n$ $1$'s and $n$ $0$'s and my question is what is the number of ways to arrange them avoiding $3$ equal consecutive numbers.
So for ...
15
votes
0answers
618 views
Prove that sum is finite
Let $j \in \mathbb{N}$. Set
$$
a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!}
$$
and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$.
Please help me to prove that the following sum is ...
14
votes
4answers
400 views
Find the infinite sum $\sum_{n=1}^{\infty}\frac{1}{2^n-1}$
How to evaluate this infinite sum?
$$\sum_{n=1}^{\infty}\frac{1}{2^n-1}$$
13
votes
3answers
406 views
Proving that $\sum_{i=0}^{n}\binom{n}{i}i^{n-i}(n-i)^{i}\le\frac{1}{2}n^n$
How can we prove that
$$\displaystyle\sum_{i=0}^{n}\binom{n}{i}i^{n-i}(n-i)^{i}\le\dfrac{1}{2}n^n$$
where $\displaystyle\binom{n}{i}=\dfrac{n!}{i!(n-i)!}$.
This inequality is very interesting. I ...
13
votes
1answer
411 views
A nice log trig integral
Show that :
$$\int_{0}^{\frac{\pi }{2}}{\frac{{{\ln }^{2}}\cos x{{\ln }^{2}}\sin x}{\cos x\sin x}}\text{d}x=\frac{1}{4}\left( 2\zeta \left( 5 \right)-\zeta \left( 2 \right)\zeta \left( 3 \right) ...
13
votes
2answers
135 views
Evaluate $\sum_{n=1}^\infty 1/n^2$ using $\int_0^1 \int_0^1 \frac{\mathrm{d}x \, \mathrm{d}y}{1-xy}$
This paper http://math.ucsb.edu/~cmart07/Evaluating%20Integrals.pdf hints at a way to compute the sum $$ \sum_{n=1}^\infty \frac{1}{n^2} $$ by expanding it into the double integral $$\int_0^1 \int_0^1 ...
12
votes
2answers
225 views
proof of $\sum\nolimits_{i = 1}^{n } {\prod\nolimits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } = 1$
i found a equation that holds for any natural number of n and any $x_i \ne x_j$ as follows:
$$\sum\limits_{i = 1}^{n } {\prod\limits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } ...
11
votes
3answers
270 views
A finite summation involving $2013$
Can you help me compute the summation below?
$$1+\frac{1}{2}+\cdots+\frac{1}{2013}+\frac{1}{1\cdot2}+\frac{1}{1\cdot3}+\cdots+\frac{1}{2012\cdot 2013}+\cdots+\frac{1}{1\cdot2\cdots2013}$$
10
votes
5answers
348 views
Is there a fundamental reason that $\int_b^a = -\int_a^b$
Is there a fundamental reason that switching the order of the limits in an integral results in the negative, i.e., $$\int_b^af(x)\,dx = -\int_a^bf(x)\,dx?$$ As far as I can tell, this is just chosen ...
10
votes
7answers
268 views
Summation simplification $\sum_{k=0}^{n} \binom{2n}{k}^2$
$\sum_{k=0}^{n} \binom{2n}{k}^2$
So i'm trying to simplify this one and I'm stuck in nowhere. Some kind of tip would be appreciated.
Thanks! :)
10
votes
2answers
82 views
Formula for finite sum
Let $n$ and $k$ be positive integers, $k\leq n$. Is there a formula for this sum?
$$
\sum_{(\alpha_{1}, \alpha_{2},\ldots, \alpha_{k}):\,\, 1\leq\alpha_{1}<\alpha_{2}<\ldots<\alpha_{k}\leq ...
10
votes
1answer
352 views
How to evaluate $\int_{0}^{1}{\frac{{{\ln }^{2}}\left( 1-x \right){{\ln }^{2}}\left( 1+x \right)}{1+x}dx}$
I want to evaluate $$\int_{0}^{1}{\frac{{{\ln }^{2}}\left( 1-x \right){{\ln }^{2}}\left( 1+x \right)}{1+x}dx}$$
I run this integral on Maple, It does converge. How we get a closed form?
Is that ...
10
votes
0answers
140 views
definite and indefinite sums and integrals
It just occurred to me that I tend to think of integrals primarily as indefinite integrals and sums primarily as definite sums. That is, when I see a definite integral, my first approach at solving it ...
