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

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0
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2answers
62 views

Find the limit of the sequence $\frac{10}{n^2}\cdot \frac{n(n+1)}{2}$

I really want to understand how to do these problems so that I can do them by myself. Please help me work out this one: Find the limit of the sequence defined by $$a_n= ...
8
votes
2answers
215 views

Convergence of the sequence $\frac{1}{e^k \sin{k}}$

Does the sequence $\frac{1}{e^k \sin{k}}$ converge? If $\sin{k}$ acts as a random variable (taking on values in $(-1, 1)$), then it seems like we should be able to prove that the sequence converges ...
3
votes
3answers
108 views

For what values of $p$ is the series $\sum_n \frac{1}{n\ln(n)^{p}}$ convergent?

The series is: $\displaystyle\sum_{n=1}^\infty \dfrac1{n (\ln n)^p}$ I don't know what to do from here since $p$ is on $\ln$. Would $p$ still have to be $> 1$ since $\ln$ is changing in terms of ...
0
votes
3answers
46 views

If $a_n\ge nb_n$ and the sequence $(b_n)$ is unbounded, then the differences $a_{n+1}-a_n$ are also unbounded

Let $(a_n)_{n\geq 1}$ and $(b_n)_{n\geq1}$ be sequences of positive numbers such that $a_n\geq n b_n$ for all $n >1$. Prove that if $(a_n)_{n\geq 1}$ is increasing and $(b_n)_{n\geq 1}$ is ...
0
votes
1answer
17 views

Almost nondecreasing sequence

Let $\{a_n\}$ be a real sequence. Suppose $$\forall j~\exists n_j:n>n_j\Rightarrow a_n-a_{n+1}<\dfrac1{2^j}$$ I wonder if there has to be a $\max_{j}\{n_{j+1}-n_j\}$ or the gaps can go to ...
4
votes
1answer
315 views

Prove or disprove that there exists a unique positive integer sequence $\{a_{n}\}$ satisfying a condition

Question: Prove or disprove: there exists a unique positive integer sequence $\{a_{n}\}$ satisfying the following condition: $\forall m\in N^{+}$, there exists a unique integer sequence ...
3
votes
2answers
61 views

Finding a product limit [duplicate]

Evaluate $L=\displaystyle\lim_{n \rightarrow \infty} \left(1+\dfrac{1}{a_1}\right) \left(1+\dfrac{1}{a_2}\right) \dots \dots \left(1+\dfrac{1}{a_n }\right)$ where $a_1=1$ and $a_n=n(1+a_{n-1}), \ ...
0
votes
2answers
32 views

why the sum of nth step is $3^{n-1}(a+b)$ more than the sum of $n-1$ step!

consider two number $a$ and $b$,we make a sequence with special process,first step we sum $a$ and $b$,now we have three numbers $a$ ,$b$ and $a+b$,now we can make two other number by summing $a$ and ...
1
vote
3answers
90 views

Existence of limit for some sequences implies existence of limit

Let $f:[0,+\infty)\rightarrow R$ and for every sequence $\{x_n\}$, such that $x_{n+1}=x_n+1$, $f(x_n)\rightarrow 0$. Does it imply that $\lim_{x\rightarrow +\infty}f(x)=0$ ? I have no idea how to ...
1
vote
1answer
178 views

Calculating the Average Number of Games Required to Reach a Theoretical True Elo Skill Rating from a given Initial Elo Rating

The USCF uses the following formula for Elo rating adjustments: $$R'=R_0+K(S-E)$$ $$E=\frac{1}{1+10^{(R_n-R_0)/400}}$$ Where $R'$ is the new rating $R_0$ is the initial rating $K$ is a ...
0
votes
2answers
62 views

Convergence of general Taylor series

For any $a,h \in \mathbb{R}$, how can we see the series, $\sum_{k=0}^{\infty} \dfrac{f^{(k)}(a)}{k!} h^k$ converges to $f(a+h)$?
3
votes
3answers
122 views

Sum of the series $\frac{2}{5\cdot10}+\frac{2\cdot6}{5\cdot10\cdot15}+\frac{2\cdot6\cdot10}{5\cdot10\cdot15\cdot20 }+\cdots$

How do I find the sum of the following infinite series: $$\frac{2}{5\cdot10}+\frac{2\cdot6}{5\cdot10\cdot15}+\frac{2\cdot6\cdot10}{5\cdot10\cdot15\cdot20 }+\cdots$$ I think the sum can be converted to ...
3
votes
1answer
104 views

Eisenstein-type series

Is the series, $$1 - 24\sum_{n = 1}^\infty \frac{q^{2n}}{(1 - q^{2n})^2}, \quad q = e^{\pi i \tau}, \quad \textbf{I}[\tau] > 0,$$ somehow related to $$E_2(q) = 1 - 24\sum_{n = 1}^\infty ...
3
votes
2answers
68 views

For what values of $x$ does the geometric series $(2+x)+(2+x)^2+(2+x)^3+\cdots$ converge?

I am stuck on this geometric series question: For what values of $x$ would the infinite geometric series $(2+x)+(2+x)^2+(2+x)^3+\cdots$ converge? Formula for series: $$S_n=\frac{a(1-r^n)}{1-r}$$ ...
5
votes
2answers
492 views

If a recursive sequence converges, must its inverse be divergent?

Suppose I have a recursive sequence $\displaystyle a_{n+1} = \frac{a_{n}}{2}$. Clearly, the sequence converges towards zero. Now, suppose I define an "inverse" sequence $\displaystyle b_{n+1} = ...
0
votes
1answer
45 views

What notation to use for a sequence of integers that end with digit 5?

I need to solve a low high school home work and I ask a question about the most correct notation. The problem is to build a set of circles with $r$ and $d$ such that $d=5, 15, 25, 35,...d_{+_1}$ and ...
3
votes
2answers
93 views

Using one limit to compute other [duplicate]

I've calculated $\lim_{n\to\infty}\dfrac{1^p+2^p+\cdots+n^p}{n^{p+1}}=\dfrac1{p+1}$ where $p\in\mathbb{N}$ fixed. I feel it should help me get this one ...
1
vote
1answer
64 views

Lerch transcendent: $\text{LerchPhi}^{(0,1,0)}\left(\frac{1}{2}, 0, 2\right)$

While messing around with something I got a result on WolframAlpha with a notation like this $$\text{LerchPhi}^{(0,1,0)}\left(\frac{1}{2}, 0, 2\right)$$ I know that ...
2
votes
1answer
102 views

$f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite

I need to prove: $f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite Now, I already have a sketch for the proof: Let $\{x_n\}$, a sequence such ...
2
votes
2answers
313 views

Find sum of $n$ terms of the series $12+14+24+58+164+\cdots$

Find sum of $n$ terms: $12+14+24+58+164+\cdots$ I have tried my best but could not proceed
1
vote
2answers
149 views

$a_1 = 2, a_{n+1} = 2a^2_n+1, a_n = ?$

I got this problem from my friend. I have been doing it for hours. $a_1 = 2$ $a_{n+1} = 2a^2_n+1$ $a_n = ?$ Could you please tell me how to solve this? Thanks! BTW: I failed to solve it by using ...
2
votes
3answers
74 views

Find partial sums of the series $12+105+1008+10011+\dots$

Find the sum of $n$ terms of this series- $$12+105+1008+10011+.....$$ I did not understand that how should I proceed with this problem.
3
votes
1answer
55 views

Uniform convergence of a sequence of polynomial logarithm

Let $P\in \Bbb{C}[X]$ of degree $d\ge 2$. For $n\in \Bbb{N}$ (include $O$). Denote by $P^n$ the $n$-th composition and $g_n: z\mapsto \frac{1}{d^n}\log(\max \{1,\vert P^n\vert\})$. Show that ...
0
votes
2answers
120 views

How to prove or disprove the convergence of the series $\sum_{n=1}^{\infty}\frac{(-1)^n\cdot a_{n}}{n}$?

Question: Assume that $a_{n}\in\mathbb R$, and let the series $$\sum_{n=1}^{\infty}a_{n}$$ be convergent I would like to prove or disprove the convergence of ...
20
votes
5answers
2k views

Where did the negative answer come from?

The question is to evaluate $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots }}}}$ $$x=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots }}}}$$ $$x^2=2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots }}}}$$ $$x^2=2+x$$ ...
14
votes
1answer
237 views

Sum the series $\sum_{n = 0}^{\infty} (-1)^{n}\{(2n + 1)^{7}\cosh((2n + 1)\pi\sqrt{3}/2)\}^{-1}$

In one of his letters to G. H. Hardy, Ramanujan gave the following sum $$\dfrac{1}{1^{7}\cosh\left(\dfrac{\pi\sqrt{3}}{2}\right)} - \dfrac{1}{3^{7}\cosh\left(\dfrac{3\pi\sqrt{3}}{2}\right)} + ...
4
votes
2answers
759 views

Prove that if an infinite series converges, then the associative property holds

I'm self-studying from the book Understanding Analysis by Stephen Abbott and have no idea how to do exercise 2.5.2 on page 57. The exercise is as follows: Prove that if an infinite series converges, ...
2
votes
1answer
50 views

Is $\sum_{n=1}^N e^{2 \pi p_n z i}$ bounded for irrational $z$?

Let $p_n$ be the $n$th prime number. If $z$ is irrational real, is it known whether the partial sums $\sum_{n=1}^N e^{2 \pi p_n z i}$ are bounded? It seems the partial sums are unbounded if $z$ is ...
3
votes
1answer
99 views

Compute the following series $\sum_{n=1}^{+\infty}\frac{1}{(n+a)(n+b)}$

Does the following series have a 'closed' form : $$\sum_{n=1}^{+\infty}\frac{1}{(n+a)(n+b)}.$$ Where $n\in \Bbb{N}$ and $a,b \in (0,+\infty)$ For $a,b$ integer we can use Partial fraction ...
3
votes
2answers
100 views

Definite integral into indefinitie series

Convert $\displaystyle \int_0^1 e^{x^2}\, dx$ to an infinite series.
0
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1answer
44 views

Does every convergent sequence have a sub-sequence whose terms comes closer than any positive sequence?

Let $(x_n)$ be convergent sequence of real numbers and $(y_n)$ be any sequence of positive real numbers , then is it true that there is a sub-sequence $(x_{r_n})$ such that ...
2
votes
3answers
104 views

Landau's proof that $\zeta(s)=\prod_{p} \frac{1}{1-p^{-s}}$

In the Handbuch, Landau proves that for all $s>1$ the following equality holds $$\zeta(s)=\prod_{p} \frac{1}{1-p^{-s}}.$$ I'm having trouble with the following part of his proof: Landau says that ...
3
votes
0answers
83 views

Generalization of small set/large set

A small set is a subset of the positive integers, such that the infinite sum of the reciprocals of the members of the set converges. Conversely, the sum of the reciprocals of a large set diverges. ...
4
votes
1answer
74 views

Is Banach space a correct context to study sequences and series?

Recently, I have reviewed elementary analysis and I realized that every theorem in the text(Rudin-PMA) about series can be generalized to Banach space. Here is an example. Below is the theorem ...
5
votes
3answers
1k views

Determining if a recursively defined sequence converges and finding its limit

Define $\lbrace x_n \rbrace$ by $$x_1=1, x_2=3; x_{n+2}=\frac{x_{n+1}+2x_{n}}{3} \text{if} \, n\ge 1$$ The instructions are to determine if this sequence exists and to find the limit if it exists. I ...
0
votes
1answer
43 views

Convergence of $\sum_{n=1}^{\infty} n$ and integral test [duplicate]

I have read online, that one can show that $\sum_{n=1}^{\infty}n = -\frac{1}{12}$. But isn't this a Riemann Series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$, where $p=-1$. And if so, can't ...
0
votes
1answer
85 views

Does it converges? $ \sum \frac{\sqrt{n}\ln(n)}{n^2+1} $

As I checked on Wolfram Alpha I know that $$ \sum \frac{\sqrt{n}\ln(n)}{n^2+1} $$ Converges. But have tried many tests to show that, without success. I tried ratio/root (inconclusive). Cauchy test ...
2
votes
3answers
168 views

Finding the second derivative of an infinite series

I'm asked to find the 2nd derivative of $$f(x)=-2x+\frac{2x^3}{3!}-\frac{2x^5}{5!}+\cdots+(-1)^{n+1}\frac{2x^{n+1}}{(2n+1)!}+\cdots=\sum \limits_{n=0}^{\infty} \frac{(-1)^{n+1}2x^{2n+1} }{(2n+1)!}$$ ...
2
votes
1answer
239 views

Complex Analysis Weierstrass M-Test

Prove that each of the following series converges uniformly on the corresponding subset of $\mathbb C$: $$\begin{align*} \text{(a)} \; & \sum_{n=1}^\infty \frac{1}{n^2 z^{2n}}, & & ...
1
vote
4answers
511 views

Finding the sum of a series till $n$ terms

Series: 5, 11, 19, 29, 41 Find the sum of the series up to $n$ terms. Well the method that comes to my mind is to find the nth term of the sequence, and then find their summation. I use the basic ...
1
vote
1answer
34 views

Recovering a continuous function from a discrete one.

Consider a well-behaved function $f(x)$ defined on $x\geq0$, and construct a discretized version of it using the Dirac-delta function: $$ ...
0
votes
1answer
54 views

$\lim_{x\rightarrow {0}^{+}}\sum_{n=1}^{\infty}{(-1)}^{n-1}\frac{1}{n!{x}^{n}}=?$

$$\lim_{x\rightarrow {0}^{+}}\sum_{n=1}^{\infty}{(-1)}^{n-1}\frac{1}{n!{x}^{n}}=?$$ I found this question during my study. In my opinion ,it is not difficult to solve ,but it is interesting. So I ...
2
votes
3answers
64 views

How to calculate the integral of $\sum_{n=1}^\infty (1/r)^{n+1} r^2$?

How to calculate this integral? $$\int\limits_0^1 {\sum\limits_{n = 1}^\infty {\left( {\frac{1}{r}} \right)} } ^{n + 1} r^2 dx$$ Here $r$ is a real number
0
votes
1answer
44 views

Power of matrix expansion

We know about the expansion $(a+b)^n\tag 1$,for scalar variables. What will be the equivalent when we want to find $(A+B)^n \tag 2 $, when A and B are square matrices? Can we treat it as same as ...
10
votes
3answers
803 views

Consecutive Prime Gap Sum (Amateur)

List of the first fifty prime gaps: 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4. My ...
4
votes
1answer
113 views

'Deriving' the Laplace Transform from the $z$ Transform: Missing a $\Delta t$

Textbooks normally give the following 'derivation' (or justification, if you prefer) of the z-Transform from the Laplace Transform. Let $x(t)$ be a signal defined on $t\geq 0$, and write a discretized ...
0
votes
1answer
60 views

Calculation of a series

Calculate the series $$\sum^{\infty}_{n=1}\frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n^2+n}}$$ I tried the root test but couldn't figure out, any ideas?
3
votes
5answers
200 views

Is the series $\sum _{n=1}^{\infty } (-1)^n / {n^2}$ convergent or absolutely convergent?

Is this series convergent or absolutely convergent? $$\sum _{n=1}^{\infty }\:(-1)^n \frac {1} {n^2}$$ Attempt: I got this using Ratio Test: $$\lim_{n \to \infty} \frac{n^2}{(n+1)^2}$$
1
vote
1answer
61 views

$\Delta^kn^\alpha$ converges monotonically to zero when $\alpha<k$

I am trying to prove that the $k$-th finite difference of the series $n^\alpha$ converges to zero monotonically as $n\to\infty$ when $\alpha<k$. The differential analogue is ...
0
votes
3answers
48 views

Question on Sequences and limits

If sequence {$a_n$} satisfies $\displaystyle \lim_{n \to \infty} (2n-1)a_n=40$, what is the value of $\displaystyle \lim_{n \to \infty}na_n$ ? Any hints ?