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.
65,953
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Interchange of limit and sum [closed]
Let $X:=(X(i))\in\ell^1(\mathbb{N})$ and $\{X_n\}$ be sequence in $\ell^1(\mathbb{N})$ such that
(i) $X_n$ converges to $X$ in sup norm,
(ii) $\sum_{i}X_n(i)$ is a Cauchy sequence.
Then is it true ...
0
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0
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15
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Comparison principle for order of convergence
Let $0< x_n < y_n$ and $y_n \rightarrow 0$ with order 1.
More precisely
\begin{align}
\frac{y_{n+1}}{y_n}=C, \text{for } 0<C<1.
\end{align}
Can we say something about the order of ...
- 534
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42
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Is the reciprocal golden ratio well approximated by this exponentially sparse series of reciprocal Fibonacci numbers?
Let $1/\phi= \phi-1\approx0.618\,$ denote the reciprocal golden ratio and $\mathrm F(k)\;(k=0,1,...)$ the Fibonacci numbers, where $\mathrm F(0)=0,\mathrm F(1)=1,$ and $\mathrm F(k+1)=\mathrm F(k)+\...
- 19k
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2
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63
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Telescopic summation for AGP: $R_n=\sum_{k=1}^n k r^{k-1}$
By the text-book method the summation of AGP is well known as: $$R_n=\sum_{k=1}^n k r^{k-1}=\frac{1-r^n-nr^n(1-r)}{(1-r)^2}.......(*)$$
We can get summation of a GP $(S_n=\sum_{k=1}^{n} r^{k-1})$ ...
- 43.6k
-2
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36
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Formula for denominators of sums ∑1/x², ∑1/x⁴, ∑1/x⁶ ... [duplicate]
Let's take this series:
$$
\sum_{x=1}^\infty{\frac{1}{x^n}}
$$
For even values of $n$, the series converges as follows:
n
Sum
2
$$\frac{\pi^2}{6}$$
4
$$\frac{\pi^4}{90}$$
6
$$\frac{\pi^6}{945}$$
...
0
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0
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47
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An example of infinite divergent series giving rational fraction of Pi.
Can an example of divergent integer sequence along some regularization method be found where the generalized sum is $c π^k $, with c, k rational (or rational complex number of the form p + qi, where p ...
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7
votes
5
answers
221
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Power series where the number $e$ is a root
I have been going at this question for weeks now and couldn't find anything.
Can we have a series of the form:
$$f(x)=\sum_{n=0}^{\infty} a_n x^n$$
where $a_n$ are rationals and not all $0$ such that $...
- 89
2
votes
2
answers
68
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Optimal strategy for uniform distribution probability game
There are 2 players, Adam and Eve, playing a game. The rules are as follows: $n$ and $d$ are chosen randomly. Adam samples a value $v$, distributed uniformly on $[0,n]$, and can either cash out $v$ or ...
- 21
1
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1
answer
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Limit of $\sqrt[n]{2-x_n}$
Let $x_n$ be the real positive root of equation: $$x^n=x^{n-1}+x^{n-2}+\ldots+x+1$$
Find $\lim{(2-x_n)^{\frac{1}{n}}}$
Here is what I tried:
Initial:
$$x^n > 1 \Rightarrow x > 1$$
It is easy to ...
- 783
0
votes
0
answers
47
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Does this recurrent sequence have a limit?
I have a sequence $a_1 = 1$, $a_2 = 0$, $$a_k = \frac{k*(k+2)+(k+1)}{k*(k+2)}(a_{k-1} -\frac{k-1}{k*(k-2) + (k-1)}a_{k-2})$$ for $k \geq 3$,
I want to know whether this sequence would converge to $0$ ...
- 165
3
votes
1
answer
67
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Example of a series where the ratio test $\limsup |a_{n+1}/a_n|$ can be applied, but $\lim |a_{n+1}/a_n|$ cannot
The ratio test asserts the absolute convergence of $\sum_{n\geq 1}a_n$ if $$\limsup \bigg |\frac{a_{n+1}}{a_n}\bigg |<1$$
In calculus, we learn the seemingly weaker form
$$\lim \bigg |\frac{a_{n+1}}...
- 1,719
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0
answers
41
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Fourier series problem... :( [closed]
How do you find the Fourier series of $f(x)=|\cos(x)|$ in $0<x<2\pi$?
1
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0
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49
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The number of ways of writing $k$ as a sum of the squares of "not so big" two elements
This question arises from the attempt to compute the Euler characteristic
of a space using a Morse function.
We fix a positive integer $n$. For each integer $k$ which satisfies the condition
$$1\leq k ...
-3
votes
0
answers
39
views
Is it ever useful in solving an equation to add the integer interpolation of a real number as a matrix product? [closed]
It is always possible to add a real number to one side of an equation without invalidating the inequality, as long as one adds an expression equal to that number to the other side of the equation. ...
- 165
2
votes
2
answers
74
views
Interesting Weighted Sum over Even Fibonacci Numbers
Doing some reading when I come across this: " ...clearly $$\sum_{n=1}^{\infty}\frac{(n+1)F_{2n}}{3^{n+1}} = 9$$ where $F_n$ is the nth Fibonacci number evaluates to $9$. We derive this solution ...
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