All Questions
47
questions
8
votes
2
answers
170
views
Let $f(\frac ab)=ab$, where $\frac ab$ is irreducible, in $\mathbb Q^+$. What is $\sum_{x\in\mathbb Q^+}\frac 1{f(x)^2}$?
Let $f(\frac ab)=ab$, where $\frac ab$ is irreducible, in $\mathbb Q^+$. What is $\sum_{x\in\mathbb Q^+}\frac 1{f(x)^2}$?
Club challenge problem. I don't think it's possible to do with only high ...
- 143
7
votes
2
answers
317
views
Stuck with this limit of a sum: $\lim _{n \to \infty} \left(\frac{a^{n}-b^{n}}{a^{n}+b^{n}}\right)$.
Here's the limit: $$\lim _{n \to \infty} \left(\frac{a^{n}-b^{n}}{a^{n}+b^{n}}\right)$$
The conditions are $b>0$ and $a>0$.
I tried this with the case that $a>b$:
$$\lim _{n \to \infty} \...
- 361
5
votes
3
answers
171
views
How to evaluate $\sum_{n=2}^\infty\frac{(-1)^n}{n^2-n}$
How would you go about evaluating:$$\sum_{n=2}^\infty\frac{(-1)^n}{n^2-n}$$
I split it up to $$\sum_{n=2}^\infty\left[(-1)^n\left(\frac{1}{n-1}-\frac{1}{n}\right)\right]$$
but I'm not sure what to ...
- 149
5
votes
1
answer
95
views
Is $f(x)=\sum_{k\in\mathbb N}\frac1k\sin\frac x{2^k}$ bounded?
$$f(x)=\sum_{k\in\mathbb N}\frac1k\sin\frac x{2^k}$$Is this function bounded?
So obviously this converges because $|\frac1k\sin\frac x{2^k}|<|\frac x{2^k}|$ and $\sum\frac x{2^k}$ converges by the ...
- 55
4
votes
2
answers
212
views
Asymptotic of a sum: $\sum_{k=n}^{\infty}{f(k)} = \int_{n}^{\infty}{f(t)dt} +\frac{f(n)}{2}+\mathcal{O}(f'(n))$
Someone told me that the following formula holds for $f$ differentiable and decreasing, with $\lim_{x\rightarrow +\infty}{f(x)}=0$.
$$\sum_{k=n}^{\infty}{f(k)} = \int_{n}^{\infty}{f(t)dt} +\frac{f(...
- 301
4
votes
1
answer
124
views
How find this sum $\sum_{k=1}^{\infty}\frac{1}{1+a_{k}}$
Let $\{a_{n}\}$ be the sequence of real numbers defined by $a_{1}=3$ and for all $n\ge 1$,
$$a_{n+1}=\dfrac{1}{2}(a^2_{n}+1)$$ Evaluate
$$\sum_{k=1}^{\infty}\dfrac{1}{1+a_{k}}$$
My idea 1:
...
- 93.6k
4
votes
2
answers
460
views
What's a closed form for $\sum_{k=0}^n\frac{1}{k+1}\sum_{r=0\\r~is~odd}^k(-1)^r{k\choose r}r^n$
I want to use a closed form of
$$\sum_{k=0}^n\frac{1}{k+1}\sum_{\ \ \ r=0\\r\text{ is odd}}^k(-1)^r{k\choose r}r^n$$
and
$$\sum_{k=0}^n\frac{1}{k+1}\sum_{\ \ \ r=0\\r\text{ is even}}^k(-1)^r{k\choose ...
- 30.1k
3
votes
2
answers
221
views
Convergence of $\sum_{n=0}^{\infty} \frac{4^n}{3^n+7^n}$
I need help with this.
$\sum_{n=0}^{\infty} \frac{4^n}{3^n+7^n}$
I know that it converges but i can not proove why.
I tried to rewrite it, it seems to be a geometric serie. I tried to do a common ...
- 47
3
votes
4
answers
192
views
How to prove that $1+\frac11(1+\frac12(1+\frac13(...(1+\frac1{n-1}(1+\frac1n))...)))=1+\frac1{1!}+\frac1{2!}+\frac1{3!}+...+\frac1{n!}$?
I'm trying to prove that
$$1+\frac11(1+\frac12(1+\frac13(...(1+\frac1{n-1}(1+\frac1n))...)))=1+\frac1{1!}+\frac1{2!}+\frac1{3!}+...+\frac1{n!}$$
Using induction, suppose that
$$1+\frac11(1+\frac12(...
user547800
3
votes
2
answers
82
views
How to find the limit:$\lim_{n\to \infty}\left(\sum_{k=n+1}^{2n}\left(2(2k)^{\frac{1}{2k}}-k^{\frac{1}{k}}\right)-n\right)$
How to find the limit:$$\lim_{n\to \infty}\left(\sum_{k=n+1}^{2n}\left(2(2k)^{\frac{1}{2k}}-k^{\frac{1}{k}}\right)-n\right)$$
I can't think of any way of this problem
Can someone to evaluate this?
...
- 1,431
3
votes
2
answers
2k
views
Partial sums of $nx^n$
WolframAlpha claims:
$$\sum_{n=0}^m n x^n = \frac{(m x - m - 1) x^{m + 1} + x}{(1 - x)^2} \tag{1}$$
I know that one can differentiate the geometric series to compute $(1)$ when it is a series, i.e. $m=...
- 11.8k
3
votes
2
answers
33k
views
Finding the infinite sum of $e^{-n}$ using integrals
I am trying to understand this:
$\displaystyle \sum_{n=1}^{\infty} e^{-n}$ using integrals, what I have though:
$= \displaystyle \lim_{m\to\infty} \sum_{n=1}^{m} e^{-n}$
$= \displaystyle \lim_{m\...
- 11.2k
3
votes
1
answer
612
views
Changing the order of summation and using a change of variables
I'm looking at the below, and I don't understand how to get the second equality using the change of variables $m=k-l$. I've tried changing the order of summation to get $$\sum_{k=2^j+1}^{2^{j+1}}\sum_{...
- 5,611
3
votes
1
answer
175
views
For which values of $p$ the series $\sum_{n = 2}^{\infty}\frac{1}{\ln^p{n}}$ converges?
I'm trying to find all values of $p$ for which the following series converges:
$$\sum_{n = 2}^{\infty}\frac{1}{\ln^p{n}}$$
So my first approach was to use the integral test because $\frac{1}{\ln^p{n}...
- 31
3
votes
0
answers
94
views
Can this summation be expressed differently?
Lets say I have a sum that states the following
$$
\sum_{j=0}^{k-c} {k-c \choose j}\ln(a)^{k-c-j} \frac{d^j}{dx^j}[(x)_c]
$$
where $(x)_c$ is the falling factorial such that
$$
(x)_c = x(x-1)(x-2)\...
- 1,957