Questions tagged [sequences-and-series]
For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.
67,622 questions
6
votes
1
answer
170
views
How fast can I guarantee convergence on a compact interval?
Suppose we have a sequence $(x_n)_n\subset [a,b]$, we know by compactness there must be at least some limit point, i.e. there exists a subsequence $n_k$ and $x\in[a,b]$ such that
$$x_{n_k}\xrightarrow{...
- 5,554
0
votes
1
answer
60
views
Behaviour of sequence defined by recurrence
I am a bit rusty on the topic, and I have been presented the following exercise I am not really sure how to deal with. We have the sequence
$$ \begin{cases} x_0 \geq 0 \\ x_{n+1} = 2|x_n -1| \end{...
- 3,764
0
votes
1
answer
57
views
Can we interchange the summations? [closed]
If the double series $\sum\limits_{k=0}^{\infty}\sum\limits_{i \in I}|c_{ki}|^2<M<\infty$, where $I$ is an uncountable set, $c_{ki}$ are complex numbers. Can we interchange the summations?
1
vote
0
answers
32
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Reduction formula or Whipple/Bailey-type transformation for a double terminating, double balanced Kampé de Fériet function $F_{1:2;2}^{1:3;3}(1,1)$
Context:
I have a double terminating, double balanced Kampé de Fériet function:
$$F_{1:2;2}^{1:3;3}\left[\left.\begin{matrix}
M:\:& A,\;B,\;C\; &;& F,\;G,\;H\\
N:\:& D,\;E\; &;&...
- 11
2
votes
2
answers
166
views
Series expansion for $\ln(k)$ for $k$ integer
I have found the following series expansion for $\ln(k)$ for $k \in \Bbb{Z}^+$ by trial and error after finding a pattern for $\ln3$, which is similar to the classic $\ln2 = \sum_{n=0}^\infty \frac{(-...
- 21
3
votes
0
answers
43
views
A characterization of bases $b$ for which the digit-sum dynamical system has only periods $\{1,2b\}$
Consider the dynamical system on $\mathbb{𝑍}_{\ge 0}^2$
$$T_b(x,y)=(y,s_b(x)+s_b(y))$$
where $s_b(n)$ is the sum of the digits of $n$ in base $b$. For a fixed base $b$, denote by $P(b)$ the set of ...
- 3,102
0
votes
1
answer
33
views
In a first countable space, cluster points of a net with a countable cofinal subset are necessarily limits of sequential subnets?
Let $X$ be a first countable topological space. Let $\phi:D\to X$ be a net such that $D$ has a countable cofinal subset $C$. Without loss of generality, we can think of $C$ to be indexed by $\mathbb N$...
- 356
2
votes
0
answers
53
views
Infinite Sums Containing Alternating Euler Sum for Odd Powers [duplicate]
When I was evaluating this monstrous integral $$
\int_0^{\frac{\pi}{2}} x^3 \ln^2 \left(\sin x\right) \, \mathrm{d}x
$$
I managed to reduce it using the fact that $$
\ln^2\left(\sin x\right) = \frac{\...
- 75
3
votes
1
answer
54
views
Writing a sequence in $c_0$ as the product of two sequences with one in $l^p$
Question. Can I write any sequence $(\mu_n)_n \in c_0$ as the products of two sequences with one of them in $l^p$? That is, can every element of $c_0$ be written as $\mu_n = a_n b_n$ such that say $(...
- 179
1
vote
1
answer
64
views
Test the convergence of a series arising from birth-death processes
Let $a_i$ and $b_i$ be nonnegative and $b_i-a_i\le -ci+d$, where $c$ and $d$ are positive. Define $$\mu_i=\frac{b_0\cdots b_{i-1}}{a_1\cdots a_i},\quad i\ge 1.$$ Does $\sum_{i=1}^{\infty}\mu_i<\...
- 155
11
votes
3
answers
2k
views
A very weird integral equation
I tried to come up with an integral equation for fun, and made this creature:$\def\d{\mathrm d}$
$$
f(x)-\int_x^{2x}f(t)\,\d t=0,
$$
so I followed these steps:
$$
\begin{aligned}
&f(x)+\int_x^{2x}...
- 111
2
votes
3
answers
165
views
Show that $\sin x\sum_{r=0}^{\infty}\frac{r!}{(2r+1)!!}\,(1+\cos x)^r=\pi-x$
For $0<x<\pi$, show that:
$$
\sin x\sum_{r=0}^{\infty}\frac{r!}{(2r+1)!!}\,(1+\cos x)^r=\pi-x.
$$
I've tried WolframAlpha and it shows
$$\sin x \sum_{r=0}^{\infty} \frac{r!}{(2r+1)!!}\,(1+\cos x)...
- 2,603
0
votes
0
answers
32
views
Convergence test for series with complex terms [closed]
Could anyone please explain how to go about checking convergence for the series $\sum_{n=1}^\infty \frac{i^n}{n-i}$? I have been able to check for absolute convergence, and the series of the absolute ...
-4
votes
2
answers
144
views
Understanding the proof of Cauchy's root test.
I am questioning a particular step of the solution presented to the following question:
Cauchy’s root test for convergence states the following: Given a series $\sum_{k=1}^\infty a_k$, define
$$\rho=\...
- 427
9
votes
3
answers
311
views
Prove that $2\int_0^1 \frac{\operatorname{Li}_2(x(1-x))}{1-x+x^2}\mathrm{d}x=\int_0^1 \frac{\ln x \ln(1-x)}{1-x+x^2}\mathrm{d}x$
I'd like to prove that
$$2\int_0^1 \frac{\operatorname{Li}_2(x(1-x))}{1-x+x^2}\mathrm{d}x=\int_0^1 \frac{\ln x \ln(1-x)}{1-x+x^2}\mathrm{d}x.$$
Ok, someone said that this holds, but I tried really ...
- 195
