All Questions
16
questions
3
votes
4
answers
2k
views
Algebric proof for the identity $n(n-1)2^{n-2}=\sum_{k=1}^n {k(k-1) {n \choose k}}$
Prove the identity:
$$n(n-1)2^{n-2}=\sum_{k=1}^n {k(k-1) {n \choose k}}$$
I tried using the binomial coefficients identity $2^n = \sum_{k=1}^n {n \choose k}$ but got stuck along the way.
- 2,646
6
votes
3
answers
1k
views
Explain why calculating this series could cause paradox?
$$\ln2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots
= (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots) - 2(\frac{1}{2} + \frac{1}{4} + \cdots)$$
$$= (1 + \frac{1}{2} + \frac{...
- 8,038
29
votes
2
answers
829
views
How to prove $\sum_{n=0}^{\infty} \frac{1}{1+n^2} = \frac{\pi+1}{2}+\frac{\pi}{e^{2\pi}-1}$
How can we prove the following
$$\sum_{n=0}^{\infty} \dfrac{1}{1+n^2} = \dfrac{\pi+1}{2}+\dfrac{\pi}{e^{2\pi}-1}$$
I tried using partial fraction and the famous result $$\sum_{n=0}^{\infty} \...
user208998
10
votes
4
answers
418
views
Evaluating $ \sum\limits_{n=1}^\infty \frac{1}{n^2 2^n} $
Evaluate
$$ \sum_{n=1}^\infty \dfrac{1}{n^2 2^n}. $$
I have tried using the Maclaurin series of $2^{-n}$ but it further complicated the question. Moreover, I have also tried taking help from another ...
user208998
2
votes
1
answer
4k
views
Find the sum of $\sin^2(n)$
I have no clue how to solve this a detailed solution would be great$$\sum_{n=1}^N \sin^2(n)=? $$
user609402
1
vote
2
answers
239
views
Formula for the general Cavalieri Sum: $S_n(p)=\sum\limits_{k=1}^{n} k^p\,\,\,n, p\in\mathbb N$ [duplicate]
What kind of formula is there that can be used for calculating the sum of power of $x$ of numbers from $1$ to $a$?
I know that the sum of numbers from $1$ to $a$ is $\ (n^2 + n)/ 2 \ $ and that the ...
- 111
23
votes
4
answers
1k
views
Calculate $\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$
I'm an eight-grader and I need help to answer this math problem.
Problem:
Calculate $$\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$$
This one is very hard for me. ...
- 809
14
votes
3
answers
1k
views
Sum of the first integer powers of $n$ up to k
Pascal's triangle has a lot of interesting patterns in it; one of which is the triangular numbers and their extensions. Mathematically:
$$\sum_{n=1}^k1=\frac{k}{1}$$
$$\sum_{n=1}^kn=\frac{k}{1}\cdot\...
- 371
3
votes
3
answers
165
views
Evaluating double sum $\sum_{k = 1}^\infty \left( \frac{(-1)^{k - 1}}{k} \sum_{n = 0}^\infty \frac{1}{k \cdot 2^n + 5}\right)$
Find $$\sum_{k = 1}^\infty \left( \frac{(-1)^{k - 1}}{k} \sum_{n = 0}^\infty \frac{1}{k \cdot 2^n + 5}\right)$$
So far, I've gotten that the sum of the left is equal to $\log(2),$ meaning we have to ...
- 551
3
votes
2
answers
607
views
Combination of quadratic and cubic series
I'm an eight-grader and I need help to answer this math problem (homework).
Problem:
Calculate $$\frac{1^2+2^2+3^2+4^2+...+1000^2}{1^3+2^3+3^3+4^3+...+1000^3}$$
Attempt:
I know how to calculate ...
- 809
2
votes
3
answers
256
views
Evaluate $\sum\limits_{r=1}^\infty(-1)^{r+1}\frac{\cos(2r-1)x}{2r-1}$
I would like to know how to evaluate $$\sum\limits_{r=1}^\infty(-1)^{r+1}\frac{\cos(2r-1)x}{2r-1}$$
There are a couple of issues I have with this. Firstly, depending on the value of $x$, it seems, at ...
- 6,560
1
vote
1
answer
83
views
How prove this identity $\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\min{\{i,j\}}(a_{i}-a_{i+1})(b_{j}-b_{j+1})=\sum\limits_{i=1}^{n}a_{i}b_{i}$?
Show that this identity;
$$\sum_{i=1}^{n}\sum_{j=1}^{n}\min{\{i,j\}}(a_{i}-a_{i+1})(b_{j}-b_{j+1})=\sum_{i=1}^{n}a_{i}b_{i}$$
where $a_{i},b_{i}\in R,(i=1,2,\cdots,n),a_{n+1}=b_{n+1}=0$
It seem can ...
- 93.6k
1
vote
1
answer
79
views
Simplify $\sum_{s=0: s \text{ even }}^\infty \sum_{m=0: \text{ even }}^\infty b_{s,m}x^s(1-x^2)^{\frac{m}{2}}$
I have the following double sum that I am trying to simplify into a single sum:
\begin{align}
\sum_{s=0}^\infty \sum_{m=0}^\infty b_{2s,2m}x^{2s}(1-x^2)^{m}
\end{align}
where $b_{s,m}$' are ...
- 2,941
0
votes
2
answers
115
views
Second-order partial derivative of $S = \sum_{k=0}^{2n}\left(\sum_{i=\max(0,k-n)}^{\min(n,k)}a_{i}a_{k-i}\right)$ [closed]
Given the following finite sum:
$S = \sum_{k=0}^{2n}\left(\sum_{i=\max(0,k-n)}^{\min(n,k)}a_{i}a_{k-i}\right)$
From this summation, I want to calculate explicitly each element $(i,j)$, i.e. the ...
- 85
0
votes
2
answers
259
views
Product of two finite sums $\left(\sum_{k=0}^{n}a_k\right) \left(\sum_{k=0}^{n}b_k\right)$
What is the product of the following summation with itself:
$$\left(\sum_{k=2}^{n}k(k-1)a_{k}t^{k-2} \right) \left(\sum_{k=2}^{n}k(k-1)a_{k}t^{k-2}\right) $$
Is the above equal to the double summation ...
- 85