For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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4
votes
1answer
71 views

Does this series converge?

I showed that, by a combination of the root test and Stirling's approximation, the series $$\sum \frac{n^n}{n!}$$ converges (the ratio test is inconclusive.) However a solution that I am comparing my ...
2
votes
1answer
76 views

Cesaro summation

Let's consider $\{a_{n} \} -$ a bounded sequence of real numbers. Is it true that $\frac{1}{n} \sum_{k=0}^{n-1}{|a_{k}|^{p}}$ (Cesaro sums) converges or diverges for all $p \geq 1$? (more presicely: ...
-7
votes
2answers
86 views

Number sequence [closed]

The following terms are found by alternately adding 4 and 6 to the previous term. The first six terms are 13, 17, 23, 27, 33, 37. (a) Find the 80th term. (b) The nth term is 203. Find n. Note: ...
1
vote
1answer
43 views

$a_{3n} = {-1}/ \sqrt[3]n, \ a_{3n+1} = {-2}/ \sqrt[3]n, \ a_{3n+2}= {3} / \sqrt[3]n$ then $\sum_{n=1}^{\infty} a_n$ converges

Let $a_n$ defined by: $$a_{3n} = \frac{-1}{\sqrt[3]n},\quad a_{3n+1} = \frac{-2}{\sqrt[3]n},\quad a_{3n+2}=\frac{3}{\sqrt[3]n} $$ show that $\sum_{n=1}^{\infty} a_n$ converges. I thought about ...
0
votes
1answer
26 views

proof of $\|y-\sum_{i=1}^n a_ix_i\|\leq \sum_{i=1}^n a_i\|y-x_i\|$

How to prove $\|y-\sum_{i=1}^n a_ix_i\|\leq \sum_{i=1}^n a_i\|y-x_i\|$? $\ \ \ \ \sum_i a_i = 1$, $0\leq a_i \leq 1$ and $y,x_i\in \mathbb{R}^m, \ \ \forall i$ It seems simple however I ...
3
votes
1answer
63 views

The growth of a cyclic sequence with $n$ terms under the rule $x_i \mapsto x_{i+1}+x_i$ and $x_{n}\mapsto x_1+x_n$

Say we have $100$ terms connected in a circle, starting with $x_1$ and going clockwise to $x_2$, to $x_3$, etc. and with $x_{100}$ going to $x_1$. Now initially $x_1 = 1$ and $x_2=-1$ and the rest of ...
0
votes
0answers
25 views

Given these points can I determine the curve?

Background Let us define a function $f(x)$ $$ \sum_{r=1}^n f(r) = g(n) $$ Now we know there must exist a $k$ such that: $$ g'(k_n) = \frac{g(n+1) - g(n)}{(n+1) -n} = f(n+1) $$ Such that $n\leq ...
0
votes
0answers
18 views

Uniform and ordinary convergence of a series.

I have a series of the kind $\sum\limits_{n=1}^{\infty} \frac{a_nt^n}{1+t^{2n}}$, $t\in(0,\infty)$. Btw, $a_n$ are real and they are determined by several integrals, but it is possible to compute any ...
0
votes
1answer
37 views

On the proof of if a sequence of function converges uniformly on $(a,b)$ then it must converge uniformly on $a$ and $b$.

I a trying to find a proof of the following: Given a sequence of functions $f_n$ that converges uniformly to $f$ on an interval $(a,b)$ where $a,b \in R$ then it converges uniformly even at $a$ and $...
3
votes
1answer
37 views

Why are the factors of some solutions to a Pell equation also a solution?

I came across this observation while trying to answer this post using the Pell equation $x^2-2y^2=1$. Define, $$P(m) = \frac{ (3+2\sqrt{2})^m+(3-2\sqrt{2})^m}{2}$$ $$Q(m) = \frac{ (3+2\sqrt{2})^m-(3-...
4
votes
1answer
266 views

Prove the identity $\cosh(2x)=\cosh^2(x)+\sinh^2(x)$ using the Cauchy product. [closed]

Prove the identity $$\cosh(2x)=\cosh^2(x)+\sinh^2(x)$$ using the Cauchy product and the Taylor series expansions of $\cosh(x)$ and $\sinh(x)$. The relations involving the exponential function are ...
6
votes
4answers
186 views

On a recursive sequence exercise.

I have the following recursive sequence of which I want to prove the convergence: $$x_{n+1} = \frac{x_n +1}{x_n +2 }$$ and $x_1 = 0$ I have proved that it is bounded above by $1$ and that it is ...
1
vote
2answers
124 views

Finding elements of cartesian product satisfying a set of constraints in GAP

Given a fixed positive integer $n \geq 3$, I want to generate a list of lists in gap with the following properties: each list has $n$ entries $a_0,a_1,\ldots,a_{n-1}$. We have: $\min(a_0,a_1,\ldots,...
4
votes
2answers
97 views

Mathematically, how does one find the value of the Ackermann function in terms of n for a given m?

Looking at the Wikipedia page, there's the table of values for small function inputs. I understand how the values are calculated by looking at the table, and how it's easy to see that 5,13,29,61,125 ...
10
votes
1answer
273 views

Closed form to an interesting series: $\sum_{n=1}^\infty \frac{1}{1+n^3}$

Intutitively, I feel that there is a closed form to $$\sum_{n=1}^\infty \frac{1}{1+n^3}$$ I don't know why but this sum has really proved difficult. Attempted manipulating a Mellin Transform on the ...
3
votes
1answer
30 views

Convergence of Polynomials in a Given Series

I've been working on a rather tough problem related to sums of infinite series. Say that I have a sum $\sum_{i=1}^{\infty} c_j$ that converges, and say that I have a polynomial $p(x)$ that has no ...
8
votes
1answer
125 views

Functions such that $\sum \frac{1}{x_n}$ diverges $\Longrightarrow \sum \frac{1}{x_nf(x_n)}$ diverge

Is there a $f : \mathbb{R}_+ \to \mathbb{R}_+$ such that : $f$ is an increasing bijective map of $\mathbb{R}_+$ into itself. For all $\displaystyle\sum_n \frac{1}{x_n}$ where $(x_n)$ is increasing ...
0
votes
1answer
20 views

Proving a sequence of partial sums is bounded

I think there's a mistake in my lecture notes but I'm always a little unsure when it comes to dealing with absolute values like this so I just want to check. We have to show that for any complex ...
1
vote
2answers
33 views

Taylor series for multivariable functions

To expend the function of multiple variables $$ f({\bf x})=f(x_1,x_2,\dots,x_n):\mathbb R^n\to\mathbb R $$ in Taylor series around $\bf 0$, we have $$ f({\bf x})=f({\bf 0})+Df({\bf 0})\cdot{\bf x}+\...
3
votes
1answer
18 views

Pointwise convergence of a sequence of functions $g_n$

Let $$g_n(x) = \begin{cases} 1 & \text{if } x = \frac{1}{n}\\ x & \text{if }x = 1, \frac{1}{2}, \frac{1}{3}, ..., \frac{1}{n-1}\\ 0 & \text{otherwise} \end{cases}$$ I am trying to figure ...
1
vote
1answer
47 views

Convergence of the series

Find the convergence and absolute convergence of the series $∑\frac{(-1)^{n+1} n}{1+n^2}$ For Absolute convergence, I found out by comparison tests, it is not absolutely convergent. But I couldnt ...
3
votes
1answer
49 views

Confusion regarding a series

I tried much but was unable to find the answer. $$f(x) = \frac{1}{3} + \frac{1 \cdot 3}{3\cdot 6} + \frac{1\cdot 3\cdot 5}{3\cdot 6\cdot 9} + \frac{1\cdot 3\cdot 5\cdot 7}{3\cdot 6\cdot 9\cdot 12} \...
3
votes
2answers
134 views

Evaluate $\prod \frac {2k - 1} {2k}$ [duplicate]

I came across to this: evaluate $$\prod_{k = 1}^{n} \frac {2k - 1} {2k}.$$ I brought it to a closed-form expression in asymptotics but I suspect my asymptotics is bad and I also want to see different ...
1
vote
1answer
30 views

Is it possible to derive a formula for any set of random numbers to turn it into a sequence?

Is it possible to derive a formula for any set of random numbers to turn it into a sequence? I recognise that it would be likely become extremely complex very quickly based on the amount of supposedly ...
1
vote
1answer
49 views

Proving a sequence is unbounded

The sequence is $(u_{n})_{n}$ for which $u_{n}=e^{n}$ The answer I have says that for any given positive real $U$, the term of index/position $n=\left \lfloor \ln(n+1) \right \rfloor + 1$ will be ...
0
votes
1answer
32 views

Sum of Bounded Series

I know that a series $\sum a_{n}$ is bounded if $\sup_{n} |\sum a_{n}| < \infty$. And a boundedness of series does not imply convergence unless its terms are monotonic. Convergence of series means ...
7
votes
5answers
80 views

Is there a notation for saying that $x_n\geq c$ from some $n$ and on?

I am looking for something like $\lim\inf x_n \geq c$, but I need that from some point and on it is $\geq c$, not just that the limit is $\geq c$.
4
votes
0answers
67 views

Convergence of a series in $R^2$ [duplicate]

For $(x,y)\in\mathbb R^2$, consider the series $$ \lim_{n→\infty}\sum_{l,k=0}^n\frac{k^2x^ky^l}{l!}. $$ Then the series converges for $(x,y)$ in $(-1,1)\times(0,\infty)$ $\mathbb R\times(-1,1)$ $(-...
10
votes
2answers
229 views

Calculate the sum of first $n$ natural numbers taken $k$ at a time

So sum of first $n$ natural numbers taken $1$ at a time is $n\cdot(n+1)/2$ but what about $2,3,\dots,k$ at a time? Is there a general formula? For example, taking 1 at a time $$\sum_{i = 1}^{n} i = \...
1
vote
1answer
43 views

Finding recurrence relation 2 - Ternary sequences. [duplicate]

for $n \ge 1$ , Let $f(n)$ be the number of ternary sequences {$0,1,2$} of length n, that the sum of their characters are even, and they have at least one show of $0$. How do i find the recurrence ...
2
votes
2answers
43 views

Proof verification: another convergent sequence proof

Note: Sorry, I posted this earlier with a glaringly obvious error - here's the improved version: The statement I'm trying to prove is: Let $ (x_n) $ be a convergent sequence and $ K \in \Bbb N $. ...
3
votes
3answers
198 views

What is $\sum_{n=1}^{\infty}\frac{1}{2^{n^2}}$ equal to?

So we know that $$ \displaystyle\sum_{n=1}^{\infty} \frac{1}{2^n} = 1 $$ Is there any information on what $$ \displaystyle\sum_{n=1}^{\infty} \frac{1}{2^{n^2}} $$ equals?
3
votes
2answers
118 views

Evaluate $\lim \limits_{n\rightarrow\infty}\sin^2(\pi\sqrt{n^2 + n})$

I'm struggling to find the limit at infinity of : $$\lim \limits_{n\rightarrow\infty}\sin^2(\pi\sqrt{n^2 + n}), n\in\Bbb N$$ I know it is $1$ but I don't understand why this is wrong : $\sin^2(\pi\...
2
votes
2answers
80 views

Computing an infinite trigonometric sum $\sum \frac{2}{n^2 \pi^2} \cos(n \pi x)\cos(n \pi y)$

Let $G(x,y) = \sum_{n=1}^\infty \frac{2}{n^2 \pi^2} \cos(n \pi x)\cos(n \pi y)$ I'm trying to compute this sum by understanding it as an integral kernel. This question comes from Dym and Mckean ...
1
vote
1answer
64 views

How do you evaluate this summation: $S=\sum\limits_{r=0}^{15} (-1)^r \frac{\binom{15}{r}}{\binom{r+3}{r}}$

Find S: $$S=\sum_{r=0}^{15} (-1)^r \frac{\binom{15}{r}}{\binom{r+3}{r}}$$ My attempt: I tried writing the summation as: $$S=3!(15!)\sum_{r=0}^{15} (-1)^r \frac{1}{(15-r)!(r+3)!}$$ and tried to ...
1
vote
2answers
131 views

Formula for $\sum \limits_{n=0}^{\infty} \frac{1}{(n+a)!}$

Is there a closed form for the infinite sum $$\sum \limits_{n=0}^{\infty} \frac{1}{(n+a)!} \mathrm{?}$$ where a is an integer greater than or equal to $0$. When $a=0$, the sum is just the series ...
4
votes
3answers
247 views

Find the distribution of the series $Z = X_1+X_2+…+X_N$

"Let $0<p=1-q<1$. Suppose that $X_1,X_2,...$ are independent Ge(q)-distributed R.V.'s and that $N \in Ge(p)$ is independent of $X_1,X_2,...$. Find the distribution of $Z=X_1+X_2+...+X_N$." I ...
1
vote
1answer
31 views

Let $M$ is a squared matrix. Find $M^n,n\in\mathbb{N}$

Let $M=XAX^{-1}$ where $ X= \begin{bmatrix} 1 & 2 \\ 2 & 3 \\ \end{bmatrix}$, $A= \begin{bmatrix} 1 & 0 \\ 2 & 1 \\ \end{...
4
votes
2answers
168 views

Prove that if $f$ is Riemann integrable on $[a,b]$ then so is $f^2$

Prove that if $f$ is Riemann integrable on $[a,b]$ then so is $f^2$. I have already proved that a function is Riemann integrable if and only if it is bounded and continuous a.e. If $f$ is bounded, ...
0
votes
5answers
83 views

Limit of the sequence $n(e^{1/n}-1)$ as $n \to \infty$

How can I evaluate $$\lim_{n\to \infty} n\left(e^{1/n} - 1\right)$$ I have tried it applying the Squeeze Theorem but I can't seem to get anywhere with that.
0
votes
1answer
54 views

Prove that if every subsequence converges to $a \in \mathbb{R}$ then the sequence converges to $a$

This question has been asked. For example: here. I am interested in a different way of proving this statement. A subsequence $(b_k)$ of a sequence $(a_n)$ is defined as $ b_k = a_{n_k}$ such that $...
6
votes
4answers
119 views

Show that every sequence in $\mathbb{R}$ has a monotone subsequence

So I would like some hints as to how to proceed on this problem as I am stuck with a particular case. I divided this proof up into 2 cases, where the sequence is convergent and the sequence is ...
-1
votes
1answer
62 views

If $a_{1}=1$ and $a_{2} = 2,$ and $a_{n}=a_{n-1}+a_{n-2},$ Then $a_{n}$ [duplicate]

The first term of a sequence is $1\;,$ and the second term is $2$ and every term is the sum of two preceding terms, Then $\bf{n^{th}}$ term is $\bf{My\ Try::}$ Let we define a Sequence $\{a_{...
5
votes
2answers
91 views

$ S_n=\sum_{k=1}^n\frac{1}{k}$ then is $S_n$ bounded?

Let , $\displaystyle S_n=\sum_{k=1}^n\frac{1}{k}$. Which of the following is TRUE ? (A) $\displaystyle S_{2^n}\ge \frac{n}{2}$ for every $n\ge 1$. (B) $S_n$ is bounded sequence. (C) $\...
7
votes
1answer
136 views

Prove that if $a_1 + a_2 + \ldots$ converges then $a_1+2a_2+4a_4+8 a_8+\ldots$ converges and $\lim na_n=0$

Let $a_1,a_2,a_3,\ldots$ be a decreasing sequence of positive numbers. Show that (a) if $a_1+a_2+\ldots$ converges then $\lim_{n\rightarrow\infty} n a_n=0$ (b) $a_1+a_2+\ldots$ converges ...
1
vote
1answer
31 views

Analysis: uniform convergence of a series

Question: Suppose $(a_k)$ and $(b_k)$ are sequences of bounded functions defined on a set S. a is defined and bounded on S, and $a_k\to a$ uniformly on S. Suppose that the functions $b_k\ge 0$ for ...
3
votes
1answer
108 views

Evaluating $\sum_{n=1}^{\infty} \frac{4(-1)^n}{1-4n^2}$

I recently found a series representation for 1 from the calculation of a Fourier series: $$1 = \frac{2}{\pi} + \sum_{n=1}^{\infty} \frac{4(-1)^n}{\pi(1-4n^2)}$$ From this, I can easily find that $$\...
0
votes
3answers
59 views

Easy convergent sequence proof

Let $(x_n)$ be a convergent sequence with $x_n > a \quad \forall \quad n \in \Bbb N $. Then $\lim_{n \to \infty} x_n \ge a $. Here's my attempt at a proof: Let $x^*$ be the limit of $(x_n)$ as $...
1
vote
1answer
48 views

Power series at $x=0$

$$\sum_{n=0}^∞n!x^{2n}$$ This power series has radius of convergence $R=0$ since the ratio test shows it diverges for all $x≠0$. But what if $x=0$? Then we have $\sum_{n=0}^\infty n! \cdot 0^{2n}$ ...
0
votes
1answer
51 views

How do I evaluate this sum :$ \sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}\log n}{n^s}$?

How do I evaluate this sum :$$ \sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}\log n}{n^s}$$ Note : In wolfram alpha it is convergent for $Re(s)>1$ .!! Thank you for any help