0
votes
1answer
53 views

Convergence of sequences such as $ B(n)=1/\sqrt{n^2+1}+\dots+ n/\sqrt{n^2+n}$

Examine the following arithmetic sequences if they converge or do not.The first one is $$ B(n)=1/\sqrt{n^2+1}+\dots+ n/\sqrt{n^2+n}$$ and the second $$C(n)=n/(n^2+1)+\dots+n/(n^2+n)$$ It was on our ...
9
votes
4answers
127 views

Compute $\lim_{n\to\infty}n^4\int_0^1\frac{x^n\ln^3x}{1+x^n}\ln(1-x)\,dx$

Compute \begin{equation} \lim_{n\to\infty}n^4\int_0^1\frac{x^n\ln^3x}{1+x^n}\ln(1-x)\,dx \end{equation} According to Wolfram Alpha, the limit is zero. I tried to make substitution ...
1
vote
4answers
159 views

Convergent or Divergent Integral

Convergent or Divergent? $$\int_0^1 \frac {dx}{(x+x^{5})^{1/2}} $$ I have problem with the fact that if we have integration from 0 to a say and a to infinity. How does this change the way we do ...
0
votes
3answers
87 views

Convergence or divergence of the integral $\int_0^1 dx/\sin x $

Is this Convergent or Divergent $$\int_0^1 \frac{1}{\sin(x)}\mathrm dx $$ So little background to see if I am solid on this topic otherwise correct me please :) To check for convergence I can look ...
0
votes
1answer
64 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 ...
4
votes
5answers
173 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: $$\lim_{n \to \infty} \frac{n^2}{(n+1)^2}$$
0
votes
2answers
37 views

How to use the Comparison Test to investigate the convergence of $\sum (\ln n)/n^\alpha$?

Let $$\sum\limits_{n=1}^\infty \frac{\ln n}{n^\alpha}, \alpha\in\Bbb{R}$$ I need to investigate the convergence of this series. I've read that since the series is positive for all $n$ then it ...
0
votes
1answer
44 views

Convergence of function

Suppose that $V(t)$ is nonnegative continuous function ($\forall t:V(t)\ge0$). $\dot{V}(t) = -|h(t)|^2 + f(t)g(t)$ $f(t)$ is a bounded and uniformly continuous function. $g(t)$ is a bounded and ...
1
vote
2answers
64 views

Convergence of infinite $ \sum (\frac{n}{n+1})^{n^2} $?

I have being trying to solve this convergence but with no success. Using the ratio test I have reached here: $$ a_n = \left( \frac{n}{n+1} \right)^{n^2} $$ And $$ \frac{1}{a_n} = \left( ...
2
votes
2answers
57 views

Verifaction of convergence/divergence exercise

I have the following assignment in my textbok: Series $\sum_{n=0}^{\infty}c_{n}3^n$ is convergent. Based on that can we conclude that the following series coverge: a) $\sum_{n=0}^{\infty}c_{n}2^n$ ...
0
votes
2answers
54 views

Converges Or Diverges: $\sum _{n=1}^{\infty }\:e^{-\sqrt{n}}$

Converges Or Diverges: Attempt: int_1^∞ e^(-sqrt(n)) dn t = -sqrt(n); dt = -dn/(2*sqrt(n)); int -2*sqrt(n)/[-2*sqrt(n)] * e^(-sqrt(n)) dn int (2t)*(e^t) dt u = 2t; du = 2 dt; dv = e^t dt; ...
2
votes
2answers
59 views

Find the first 5 coefficients of the series $\frac{6x}{x+9} = \sum_{n=0}^\infty C_n x^n$

I rewrote the equation series as $$ \frac69 \sum_{n=0}^\infty \left(\frac{-1}{9}\right)^n x^{n+1} $$ And therefore have coefficients of $C_0 = 6/9, C_1 = \left( 6/9 \right) ...
4
votes
2answers
87 views

Convergence of tetration sequence.

This question arose from here. I am interested to find a nice proof about the convergence of $${^n}a=\underbrace{a^{a^{\ .^{\ .^{\ .^a}}}}}_{n\ \text{times}}.$$ I find with google a necessary and ...
0
votes
0answers
41 views

Why there's no articles about the eta function convergence?

I've been searching about a proof that the eta function converges for $\mbox{Re}(z)>0$ but the ONLY page I've found that claims to prove it was in this question: ...
5
votes
1answer
42 views

how to determine the existence of double limit?

Let $f(x,y)$ be a function of two variables. Are there any criterions to determine the existence of double limit $$ \lim_{(x,y)\to(x_0,y_0)} f(x,y)? $$ If for all $y\in(y_0-\delta,y_0+\delta)$, ...
2
votes
1answer
40 views

direction limits and double limit

Let $f(x,y)$ be a function of two variables. What is the counterexample that there exists $A$ s.t. for all $\theta$, $$\lim_{r\to 0+}f(r\cos \theta,r\sin \theta)=A$$ but double limit $$ ...
1
vote
2answers
59 views

How to prove that the sequence is decreasing $a_{n}=\frac{ln(n)}{n^2}$

Is my way/proof good and completely mathematically rigorous? $a_{n}=\frac{ln(n)}{n^2}$ --> $a_{n+1}=\frac{ln(n+1)}{(n+1)^2}$ $\frac{ln(n)}{n^2} > \frac{ln(n+1)}{(n+1)^2}$ ...
3
votes
5answers
133 views

Convergent or divergent $\sum_{n=1}^{\infty}\frac{1}{\sqrt{n^2+1}}\left(\frac{n}{n+1}\right)^n$?

Any suggestions? I have tried using D'Alembert's test, but on the end I get 1. I can't think of any other series with which to compare it. In my textbook the give the following solution which I don't ...
1
vote
3answers
112 views

Decide convergence divergence of $\sum \dfrac{1}{(\ln n)^{\ln n}}$ [duplicate]

Does the series $\sum \dfrac{1}{(\ln n)^{\ln n}}$ converges? I can intuitively say that it converges, because $(\ln n)^{\ln n} $ is going to $\infty$ on a hayabusa
0
votes
2answers
62 views

Find a sum of $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{ch(n)}{3^n}$

Find a sum of $$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{ch(n)}{3^n}$$ Could you give some some hint or some way to start this? I have tried representing ch(n) through its definition with e, but I ...
3
votes
3answers
106 views

Does $\sum_{n=1}^{\infty}\frac{n-1}{n^2}$ converge or diverge?

Is my logic OK? $a_{n}=\frac{n-1}{n^2}$ $\frac{1}{n} \leq b_{n}=\frac{n-\frac{n}{2}}{n^2}=\frac{n}{2n^2}=\frac{1}{2n} \leq a_{n}=\frac{n-1}{n^2}$ and there for the initial series diverges.
3
votes
1answer
32 views

Prove that $(x_n)_{n\geq1}$ is an arithmetic progression

Let $(x_n)_{n\geq1}$ be a sequence of integers. Define $y_n=\frac{x_n}{n},n\geq1$. The sequence $(y_n)_{n\geq1}$ is convergent and $n$ divides the sum of any $n$ consecutive terms of the sequence ...
1
vote
4answers
61 views

Why does $\sum_{k=1}^{\infty} \frac {(-1)^{k-1}}{k}$ converge conditionally?

Why does it converge conditionally? $$\sum_{k=1}^{\infty} \frac {(-1)^{k-1}}{k}$$
2
votes
4answers
53 views

How to apply the alternating series test to the series $\sum (-1)^{n+1} n/2^n$?

So, I need to test the following series for convergence or divergence: $$\sum_{n=1}^\infty (-1)^{n+1}{n\over {2^n}}$$ I know that when you use the Alternating Series Test, the series must satisfy ...
1
vote
3answers
49 views

Is the following series converging or diverging. $\sum_{n=1}^{\infty}\dfrac{n+4^n}{n+6^n}$

I know one solution. That is by Doing comparison with $\dfrac{4^n+4^n}{6^n}$ Wondering if there are more ways to do it
0
votes
2answers
39 views

Problem with proof that positive infinite series are commuative

Proof from real analysis book: Let $\sum_{n=0}^{\infty}a_n$ converge, where $a_n \geq 0, n \in \mathbb{N}$. Then the series $$ \sum_{k=1}^{\infty}a'_k = a'_1 + a'_2 + \cdots + a'_k + \cdots $$ ...
0
votes
1answer
35 views

Simpler way of proving series convergence?

Determine whether the following series converges $$\sum_{n=1}^{\infty} \left (\frac{n^4}{n^4 + 2}\right)^{n^5-3}.$$ I've found convergence using the root criterion in the following way. $\sqrt[n]{ ...
1
vote
1answer
42 views

Finding radius of convergence using root test

Find the radius of convergence of the following power series $$\sum_{n=1}^{\infty} \frac{2^n + 1}{n} x^n.$$ Using the ratio test, I have found that the radius of convergence is $R = \frac{1}{2}$. I ...
4
votes
2answers
156 views

Convergence of ${\large\int}_{-\infty}^\infty J_0(x)\,J_0(x+a)\,dx$

Consider $$I(a)={\int}_{-\infty}^\infty J_0(x)\,J_0(x+a)\,dx,$$ where $J_0(z)$ is the Bessel Function of the $1^{st}$ kind and $a>0$. Does this integral converge for any values of $a$? If so, is ...
3
votes
0answers
27 views

Convergence of sum of antiderivative and derivative

This question is inspired by this question: Solutions for $ \frac{dy}{dx}=y $?. It makes me wonder if there are any function where the sum of all antiderivative and derivative converges. The ...
1
vote
1answer
31 views

evaluating a sum using Cauchy condensation test

Let $$\sum\limits_{n\ge1}{\frac{(-1)^n}{n^\alpha \ln n}}$$ I want to check if the sum is converges absolutely. Hence, we need to check the convergence of $$\sum\limits_{n\ge1}{\frac{1}{n^\alpha \ln ...
5
votes
4answers
114 views

Is $\sum\limits_{n=1}^\infty \sin{\frac{(-1)^{n+1}}{n}}$ convergent?

$$ \mbox{Is}\quad \sum_{n=1}^\infty \sin\left(\left[-1\right]^{n + 1} \over n\right) \quad\mbox{ convergent ?.} $$ $$ \mbox{I know that }\quad\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}\quad \mbox{is ...
2
votes
2answers
63 views

Does this series violate the decreasing condition of the Integral Test for Convergence?

I'm working on the section involving the Integral Test for Convergence in my calculus II class right now, and I've run into a seeming conflict between the definition of the Integral Test, and the ...
1
vote
0answers
32 views

numerical solution of integral equation

Consider the basic type of integral equation. In particular, a volterra integral equation of the first kind. That is, we have the following integral equation $$\int_a^xf(s)g(s,x)~ds=h(x)$$ where $h$ ...
3
votes
2answers
251 views

Sequence problem

I have a calculus final two days from now and we have a test example. There's a sequence question I can't seem to solve and hope someone here will be able to help. With $a_1$ not given, what are the ...
4
votes
4answers
153 views

Convergence of $\left(1+\frac{1}{n}\right)^n$, given that $\left(1+\frac{1}{2^n}\right)^{2^n}$ converges

I know how to prove that the sequence $$\left(1+\frac{1}{2^n}\right)^{2^n}$$ has a limit. Can I use this knowledge to quickly get the fact that $$\left(1+\frac{1}{n}\right)^n$$ also has a limit?
2
votes
5answers
173 views

Is integral convergent?

I have a problem with following integral: $\int_1^\infty \frac{\sqrt{x}}{1+x} \sin(2x)dx$ I was trying to prove convergence (or divergence) of this integral, however without any success. My best ...
1
vote
1answer
66 views

Dirichlet's function

How can we see that Dirichlet's function $$D(x):=\lim_{m\to\infty} \lim_{n\to\infty} \cos^{2n}(m!\pi x)= \begin{cases} 1 & x\in\mathbb Q\\ 0 & x\notin\mathbb Q\\ ...
0
votes
0answers
30 views

A function is discontinuous at all rational points and continuous at all irrational points [duplicate]

Define $f(x)$ for $x\in[0,1]$ by $f(\frac pq)=\frac1q$ if $p$ and $q$ are relatively prime, and $f(x)=0$ if $x$ is irrational. How can we see that $f$ is discontinuous at all rational points and ...
1
vote
0answers
36 views

Fixed point method where the derivative is one - does it converge

I'm trying to see if the iterative method $x_n=g(x_{n-1})$ where $g(x)=2\sqrt{x-1}$ will converge to $2$, if I take $x_0$ that is sufficiently close to $2$. Indeed notice that $g(2)=2$. and we have a ...
0
votes
4answers
63 views

Decide convergence of this series

How to prove the series $$\sum_{n=1}^\infty \frac {e^n\cdot n!}{n^n}$$ diverges? I tried D'Alambert and result is 1 and I'm stuck with Raabe.
0
votes
1answer
43 views

What am I doing wrong in this continuity check?

I want to show that the function $f$ is discontiunous. $f$ is defined as follows: $$f(x,y) := \begin{cases} (x^2+y^2)\cdot\sin(\frac{1}{\sqrt{x^2+y^2}}) & , (x, y) \neq (0,0) \\ 0 ...
4
votes
2answers
91 views

L. Kronecker's theorem for sequences and series: $\lim_{n\to\infty}\frac{1}{b_n}\sum_{k=1}^na_kb_k=0$

Assume $\sum a_i$ is a convergent series and $b_1,b_2,\dots$ is a divergent monotonically increasing sequence. How can we see that $$\lim_{n\to\infty}\frac{1}{b_n}\sum_{k=1}^na_kb_k=0$$ Attempt: We ...
0
votes
1answer
43 views

Prove or disprove the convergence of…

I need help with the following problem, please help. For positive real x. Let $${ B }_{ n }(x)\quad =\quad { 1 }^{ x }+{ 2 }^{ x }+{ 3 }^{ x }+...+{ n }^{ x }$$ Prove or disprove the convergence ...
0
votes
1answer
45 views

Cesaro summability together with $\lim nu_n\to 0$ implies convergence

Assume the series $\sum u_n$ is Cesaro summable and $\lim_{n\to\infty} nu_n\to 0$. We want to see that the series is (Cauchy) convergent. Attempt: Let $s_n=\sum_{i=1}^n u_n$ denote the $n$-th partial ...
0
votes
0answers
32 views

Convergence of two power series

I just wanted to know, whether my results are correct. I should find the radius of convergency in both cases: $\sum_{n=1}^\infty \frac{z^{2n}}{n^23^{n}}$ with a quotient criterion ...
2
votes
4answers
94 views

How to prove this integral converge?

$$\int_{1}^{\infty }\frac{\ln x}{1+x^2}\,{\rm d}x$$ So far i tried to use the comparison test with $\int_{1}^{\infty }\frac{\sqrt{x}}{1+x^2}$ but i noticed that it's not always true. any ideas?
0
votes
4answers
43 views

What is $f$? Finding where a function converges pointwise?

I have a question. Let $f_n(x) = x^{4n} + \frac1{n^2}$. AS $n \to \infty$, $f_n$ converges pointwise to a function $f$ on $[0,1]$ What is $f$? Now if I am understanding correctly, couldn't ...
0
votes
3answers
64 views

Convergence of sequence method, Math behind intuition

Now I want to find convergence of a sequence: $$ \lim_{n \to \infty} \sqrt[n]{4^n + 5^n}$$ Now I am pretty sure I have solved this using logic on inspection: $4^n \ll 5^n$ as $n\rightarrow\infty$, ...
0
votes
1answer
36 views

convergence/divergence $\sum_{n=2}^{\infty}(\frac{n}{-n+1})^n$

I am stuck with this series $\sum_{n=2}^{\infty}(\frac{n}{-n+1})^n$. I used nth-root test, but the limit was $1$. Then I tried to think about it as $(-1)^n(\frac{n}{n-1})^n$ to use Leibniz, and I got ...