Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: recurrence relations, convergence tests, identifying sequences, identifying terms. For questions on finite sums, use the (summation) tag instead.

44,379 questions
30 views

Local isolation of zeros complex analysis

I've been studying from Basic complex analysis by Mardsen and Hofmann, and I came across the proposition and corollary on page 212: Proposition 3.2.9 Suppose $f: \Omega \rightarrow \mathbb{C}$ is ...
46 views

prove $\sum_1^\infty (-1)^k / \sqrt{k}$ converges [closed]

How could one prove that $$\sum_1^\infty (-1)^k / \sqrt{k}$$ converges?
814 views

19 views

Find first two terms of sequence where the relationship of the first three terms is known

The sequence $u_n$ is defined by: $u_{n+2}=2*u_{n+1}+2*u_n$ for $n\geq2$ with given $u_0$ and $u_1$ $u_0$ = 3 $u_1$ = 3 My goal is to find $u_2$ and $u_3$ but I'm struggling to see how I can go ...
41 views

17 views

Dirichlet series on positive reals [closed]

I consider a séquence $(a_{n})$ of positive real numbers, and $f:s \rightarrow \sum_{n \geq 1} \frac{a_{n}}{n^{s}}$. The serie converges absolutely for $\Re(s) \geqslant 0$ and the function f has a ...
17 views

Algorithm for evaluating a sequence while removing from that sequence?

Introduction : I'm looking at this problem and I've gotten stuck on it. The rules are as follows: Problem Definition : You have a sequence of numbers -- 1 through x (where x > 1). Two numbers are ...
69 views

43 views

$\sum_{n=1}^\infty \frac{a_1+a_2+\cdots+a_n}{n}$ converges/divereges?

Let $a_n\geq 0$. $\sum_{n=1}^\infty \frac{a_1+a_2+\cdots+a_n}{n}$ converges/divereges? Take $a_n=1/n$ for example, then the proposed series looks like $\sum\frac{\ln n}{n}$, which is divergent. What ...
39 views

17 views

Finite representation-infinite rings

Rings are not necessarily commutative, but associate and unital here. Recall that representation-infinite means that there are infinite non-isomorphic indecomposable modules. For a natural number $m$ ...
46 views

On a sum of infinite series

(Second image is answer) Could somebody show how exactly the last line is derived? and why are the indices on the partial sums 1 and 3? Why not some other numbers? I really don't understand..
260 views

Why do we have $(1+\frac{c}{n}+o(\frac{1}{n}))^n \to e^c$ as $n \to \infty?$ [duplicate]
We have $(1+\frac{c}{n})^n \to e^c$ as $n \to \infty$. Why does a term in $o(\frac{1}{n})$ does not change this asymptotic?
I have worked this problem at least 30 times and am still not getting the correct answer. Can anyone tell me where I'm wrong? Partial sum of a geometric series: enter image description here \sum_{...