All Questions
104
questions
2
votes
3
answers
60
views
For which values the following summation converges
I have the following problem:
For which $a$ and $b$ values does the following summation converges:
$$\sum_{n=1}^{\infty}(-1)^{n-1}\left(a-\frac{(n-b)^n}{n^n}\right)$$
I tried to solve this in many ...
- 21
5
votes
1
answer
409
views
Dini's theorem (specific case)
Note: I asked this question before but it wasn't well written, So I deleted my previous question and re-wrote it.
According to Dini's theorem:
If $X$ is a compact topological space, and $\{ f_n \}$...
- 421
2
votes
2
answers
367
views
Cauchy product summation converges
I had a previous question here, which I'm quoting:
How can I prove that the following summation converges?
$$\sum_{n=0}^\infty \sum_{k=0}^n \frac{(-1)^n}{(k+1) (n-k+1)}$$
I tried to prove ...
- 421
0
votes
2
answers
31
views
$\sum_{\alpha=0}^A\binom{A}{\alpha}(zb)^{\alpha} c^{A-\alpha} = (bz+c)^A$
Why the following equality holds:$$\sum_{\alpha=0}^A\binom{A}{\alpha}(zb)^{\alpha} c^{A-\alpha} = (bz+c)^A$$
How to prove it?
- 53
0
votes
1
answer
326
views
Dini's theorem, under which conditions?
According to Dini's theorem:
If $X$ is a compact topological space, and $\{ f_n \}$ is a monotonically
increasing sequence (meaning $f_n(x) \leq f_{n+1}(x)$ for all $n$ and $x$) of
continuous ...
- 421
0
votes
0
answers
43
views
How to prove that a non-positive series converges?
Given a(n) a series that is monotonic and converges to zero which means: a(n)---->0.
How to prove that the following summation converges for any K (a normal number):
$$\sum\limits_{n=1}^\infty (-1)^{...
- 421
0
votes
2
answers
37
views
Summation with two costs
I need to find for which alpha and beta values, the following summation will converge and for which it will diverge
$$\sum\limits_{n=1}^\infty (-1)^{n-1}\left(\alpha-\frac{(n-\beta)^n}{n^n}\right)$$
...
- 421
3
votes
3
answers
120
views
radius of convergence of a given series
How could I find the radius of convergence for the following power series:
$$
\sum\limits_{n = 1}^\infty {\frac{{2^{n - 1} x^{2n - 1} }}{{(4n - 3)^2 }}} .
$$
From what I read I need to find: $$
\...
user788860
0
votes
0
answers
37
views
Why the sum$\sum_{n=1}^\infty \left(n^{n^a}-1\right)$ converges when a is strictly less then -1
$$\sum_{n=1}^\infty \left(n^{n^a}-1\right)$$
Plugging in numbers on wolfram alpha I know that this sum converges when a is strictly less than -1. It is kinda very similar to the p-series test because ...
- 1
1
vote
3
answers
196
views
Testing a Series for Convergence or Divergence
The problem is to test the following series for convergence:
$\displaystyle\sum\limits_{i=1}^{\infty} ((i^2+1)/(i^3+1))$
I tried several tests like the root and limit comparison test but they ...
- 115
2
votes
0
answers
137
views
Can this summation be done without calculator?
Is it possible to perform the summation ,
$$\sum_{i=1}^{\infty} \frac{1}{i^i}$$
without the use of calculator?
It does converge to a finite value = 1.29129...
Wolfram Alpha link to this
Describe the ...
- 1,663
4
votes
5
answers
386
views
Show that $\sum_{n=0}^\infty \frac{1}{n+1} \binom{2n}{n} \frac{1}{2^{2n+1}} = 1.$
Question: Show that
$$\sum_{n=0}^\infty \frac{1}{n+1} \binom{2n}{n} \frac{1}{2^{2n+1}} = 1.$$
From Wolfram alpha, it seems that the equality above is indeed correct.
But I do not know how to ...
- 15.9k
3
votes
3
answers
74
views
Why would this series Absolutely Converge using Root Test?
So I was working on this problem and I got diverge, since my answer was greater than 1. The Limit was > 1, using the root test.
$$\sum\limits_{n=4}^\infty (1 +\frac{1}{n})^{-n^2}$$
I ended up with
...
- 211
2
votes
3
answers
5k
views
How to evaluate whether summation of $e^n/(ne^n+1)$ diverges or converges?
How to test this summation for divergence or convergence?
$$\sum_{n=0}^\infty \frac{e^n}{ne^n+1}$$
Edit: Here is my work, but I got it wrong. I tried using the comparison test.
\begin{align*}
a_n &...
- 29
1
vote
1
answer
219
views
Baby Rudin ex. 3.8 proof verification
The question asks: if $\sum a_n$ converges, $\{b_n\}$ is monotonic and bounded, prove that $\sum a_nb_n$ converges.
My proof goes as follows:
Let $\varepsilon>0$, and let $S_k$ denote $k$-th ...
- 13