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.
65,957
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Does there exist a positive sequence with these properties?
Let $\{x\}$ denote the fractional part of $x$ and $d_n=\text{lcm}(1,2,...,n)$. Does there exist a sequence with all positive terms $(a_n)_{n\geq 1}$ such that $$\lim_{n\to\infty} \{a_n\}=1\ \text{,}\ \...
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1
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Optimal strategy for uniform distribution probability game
There are 2 players, Adam and Eve, playing a game. The rules are as follows: $n$ and $d$ are chosen randomly. Adam samples a value $v$, distributed uniformly on $[0,n]$, and can either cash out $v$ or ...
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Where does $x\mapsto e^{-x}-(1-\frac xn)^n$ reach its maximum? [closed]
For all integer $n\ge2$, the map $f_n:[0,n]\to\mathbb{R},x\mapsto e^{-x}-(1-\frac xn)^n$ reaches its maximum at some $x_n$. I could prove that $1<x_n<2$.
I would like to show that the sequence $(...
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Limit of $\sqrt[n]{2-x_n}$
Let $x_n$ be the real positive root of equation: $$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$$
It is easy to ...
- 771
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Limit of recurent sequence with variant coefficients
I have a sequence $a_1 = 1$, $a_2 = 0$, $$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})$$ for $k \geq 3$,
I want to know whether this sequence would converge to $0$ ...
- 165
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Example of a series where the ratio test $\limsup |a_{n+1}/a_n|$ can be applied, but $\lim |a_{n+1}/a_n|$ cannot
The ratio test asserts the absolute convergence of $\sum_{n\geq 1}a_n$ if $$\limsup \bigg |\frac{a_{n+1}}{a_n}\bigg |<1$$
In calculus, we learn the seemingly weaker form
$$\lim \bigg |\frac{a_{n+1}}...
- 1,729
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Fourier series problem... :( [closed]
How do you find the Fourier series of $f(x)=|\cos(x)|$ in $0<x<2\pi$?
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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
$$1\leq k ...
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Is it ever useful in solving an equation to add the integer interpolation of a real number as a matrix product? [closed]
It is always possible to add a real number to one side of an equation without invalidating the inequality, as long as one adds an expression equal to that number to the other side of the equation. ...
- 165
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Interesting Weighted Sum over Even Fibonacci Numbers
Doing some reading when I come across this: " ...clearly $$\sum_{n=1}^{\infty}\frac{(n+1)F_{2n}}{3^{n+1}} = 9$$ where $F_n$ is the nth Fibonacci number evaluates to $9$. We derive this solution ...
- 33
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Convergence rate of Laguerre coefficients for polynomially bounded functions
Suppose $f:[0,\infty)\rightarrow\mathbb{R}$ satisfies:
$$f(x)= \sum_{n=0}^\infty \hat{f}_n L_n(x),$$
for some $\hat{f}_0,\hat{f}_1,\dots\in\mathbb{R}$, where $L_n$ is the $n$th Laguerre polynomial for ...
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Limit of $\sum_{k=1}^n \frac{1}{k+1/2} - \ln(n+1/2)$
I am trying to understand the limit of the following expression as $n \to \infty$:
$$\sum_{k=1}^n \frac{1}{k+1/2} - \ln(n+1/2)$$
Numerical investigation suggests that the limit of this expression as $...
- 105
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Sequence calculation
Is there a way to calculate each element of these sequences individually without having to calculate the previous element or sequence?
...
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Informations about a sequence from tail behaviour
Suppose $\{c_n\}_n$ is a sequence of non negative reals. We have the following three informations about it.
(a) $\sum_{k \ge n}c_k \sim \frac{1}{2n}$
(b) $\sum_{k=2}^n \frac{kc_k}{\log n} \to \frac{1}{...
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If $\forall n,\sum_ka_{n,k}^2<\infty$ and $\forall k,a_{n,k}\to b_k$, how to show that $\sum_kb_k^2<\infty$? [closed]
Let $\ell^2$ denote the metric space of all the square-summable sequences of real numbers. Let $p_n = \left( a_{n1}, a_{n2}, a_{n3}, \ldots \right)$ for $n = 1, 2, 3, \ldots$ be a sequence of points ...
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