All Questions
Tagged with divergent-series limits-and-convergence
10 questions
5
votes
0
answers
273
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$\sum_n a_n/n$, $\sum_n a_n/n^\rho$, $\sum_n a_n$… Tauberian theorems?
In analytic number theory, it is common to prove that $$\sum_{n\leq N} \frac{a_n}{n} = o(\log N)\tag{$\star$}\label{476699_star}$$ for some sequence $\{a_n\}_{n=1}^\infty$, $a_n\in \mathbb{C}$. (It is ...
- 20.2k
2
votes
2
answers
260
views
Laurent Series $\sum_{n=-1}^\infty a_n x^n$ when $a_{-1} = \infty$
When dealing with complex functions, if $f(x)$ has a simple pole, then we can find the coefficient $a_{-1}$ in the Laurent expansion $f(x) = \sum_{n=-1}^\infty a_n x^n$ by evaluating the limit $\lim_{...
- 1,730
-2
votes
1
answer
217
views
Convergence and roots of alternating periodic infinite series
Let $0<\alpha <1$ and $\beta > 0$. Consider the mapping $$F(\alpha, \beta) = \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}.$$ Can we prove $F(...
- 317
1
vote
1
answer
2k
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Interchange summation order in the limit of number of elements going to $\infty$
Considering the sum $\sum_{i=0}^{\infty}\sum_{j=0}^{\infty} a_{ij}$, in general we are not allowed to interchange the summation order (i.e. pass to $\sum_{j=0}^{\infty}\sum_{i=0}^{\infty} a_{ij}$) but ...
- 305
3
votes
1
answer
285
views
Is there an asymptotic bound between converging and diverging series? [closed]
Let us define for every $k\in\mathbb{N}$ and every large enough $x\in \mathbb{R}$,
$$
\log^{[k]}(x) =
\begin{cases}
\log^{[k-1]}(\log(x)) & k>0 \\
x & k=0
\end{cases}.
$$
It is well known, ...
- 41
4
votes
1
answer
2k
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Exchanging series and integrals
I know that I can use Lebesgue or monotone convergence theorem to exchange limit of partial sums and a Lebesgue integral, given a power series or a generic function series. But in general given a ...
- 91
1
vote
0
answers
59
views
Finding a variable P for which a sum converges [closed]
I need guidance in finding a variable P for which $ \sum _{n=4}^{\infty }\:\left(\frac{n\ln \left(n\right)-n}{\ln \left(n!\right)}\right)^p $ converges, or proof that there doesn't exist such P ...
- 11
1
vote
2
answers
604
views
Generalized limits
Cross-posted from Math SE.
The linked question explores the concept of a "generalized limit" that assigns values to sequences which diverge in the Cauchy sense. It asks the following question:
...
- 2,203
0
votes
1
answer
220
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What is a sufficient condition for summability of formel power series? [closed]
There are several kind of summability , i accrossed differents conditions for applying for example Borel summation or laplace transform which let me mixed and confused , really i don't know if i have ...
user119110
2
votes
1
answer
305
views
Division methods for divergent continued fractions
I hadn't even noticed before entering the subject above the parallel with the title of an earlier question posted here titled Summation methods for divergent series. And it's just mutatis mutandis as ...
