0
votes
0answers
18 views

Sequence of integers in arithmetic progression and convergent sequence

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
60 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
49 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
48 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
37 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
31 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
41 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
136 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
26 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 ...
0
votes
0answers
32 views

Convergence of a series using limit test

As per my understanding: When we take $\lim_{n \rightarrow \infty}$ of a function, it should approach a finite number, it converge. And if the opposite is true, it diverges. Now, the test for ...
1
vote
1answer
30 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
109 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
61 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
30 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
249 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
151 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
172 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
64 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
35 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
62 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
41 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
89 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
41 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
43 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
91 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
60 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
35 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 ...
1
vote
0answers
34 views

Convergence of $\sum_{n=1}^{\infty}{r^n \cdot \sqrt n \cdot \arctan{\frac{1}{n+1}}}$

I have to find for which $r\in\mathbb{R}$ a) series diverge b)converge absolutely c)converge not absolutely. $$\sum_{n=1}^{\infty}{r^n \cdot \sqrt n \cdot \arctan{\frac{1}{n+1}}}$$ I don't know ...
2
votes
3answers
46 views

$\lim_{n \rightarrow \infty } {n \choose k} a^n = 0$

Let $k$ a fixed positive integer and $0<a<1$ a real number. Prove that $\lim_{n \rightarrow \infty } \left( \frac{n!}{(n-k)!k!}\right) a^n = 0$ . I stuck in this limit i remember that ${n ...
1
vote
2answers
71 views

Convergence of $\displaystyle\int\frac{1}{\sqrt[3]{1-x^3}}\ dx$

Please help me to prove that this integral converges. $$\int_{0}^1 \frac{1}{\sqrt[3]{1-x^3}}\ dx $$ No ideas. Tried to find function which is bigger and converges, equivalent fun-s, but no result ...
0
votes
1answer
21 views

Subsequence to infinity proof

How to prove this: Every not bounded above sequence has subsequence which limit is infinity when n->infinity It's nearly the same what Bolzano–Weierstrass ...
0
votes
1answer
24 views

Show convergence of sequence

Consider a sequence $(x_n)$ such that $\forall n \in \mathbb N, |x_{n+1}-x_{n}|\leq 2^{-n}$ How to show that it converges? Any hints would be appreciated
1
vote
1answer
42 views

Show that this difficult sequence converge

$x_1=\sqrt{2},\,x_2=\sqrt{2+\sqrt{2}},\, x_3=\sqrt{2+\sqrt{2+\sqrt{2}}},\, \cdots$ How to show that the sequence converges? We can say that it is monotonic but if it is bounded?
1
vote
2answers
38 views

Arc length integration

Find the length of the arc formed by $x^2=10y^3$ from point A to point B, where A=(0,0) and B=(100,10). My attempt: $\int_0^{100} \! \sqrt{1+(\frac{2}{3x})^2} \, \mathrm{d}x. $ However this integral ...
0
votes
1answer
41 views

The power series $\sum_0^\infty2^{-n}z^{2n}$ converges, if

The power series $\sum_0^\infty2^{-n}z^{2n}$ converges, if a) |$z$| $\le$ 2. b) |$z$| $<$ 2. c) |$z$| $\le$ $\sqrt{2}$. d) |$z$| $<$ $\sqrt{2}$. Please anyone give me the answer. I think ...
0
votes
4answers
51 views

Does this sequence converge

Does the sequence $x_n=3^n−2^n$ converge? I can show that it is increasing but how to show that it is bounded?
1
vote
1answer
71 views

Does this converge? $\sum_{n=3}^\infty \frac1{n+ \log(n)}$

This seems too easy, my friend said he couldn't get it. maybe I am wrong?? $$\sum \limits_{n=3}^\infty \frac1{n+ \log(n)} \leq \frac1{n+n}=\frac1{2n} \leq \frac1n$$ Which converges as harmonic?
0
votes
0answers
50 views

My first integral test, is it correct?

I want to test for convergence on $\sum \limits_{n=3}^\infty \cfrac1{n(\log n)(\log(\log n))}$ Now I have just learnt the integral test off of a fellow stack exchange user(M. Vinay). $\int_3^\infty ...
1
vote
2answers
34 views

I want to prove a series converges absolutely

I want to show that: $\sum \limits_{n=1}^\infty \cfrac{n^2 + \cos(n)}{e^{n^3}}$ converges absolutely. Now, here is what I have done: $\sum \limits_{n=1}^\infty \cfrac{n^2 + \cos(n)}{e^{n^3}} \leq ...
3
votes
1answer
53 views

What is the proper way to handle the limit with little-$o$?

I was hoping to show that $$\left(1-\frac{x}{n}+o\left(\frac{2x}{n}\right)\right)^n \xrightarrow{n\to\infty} e^{-x}$$ which would be just fine without the little-$o$. Trying binomial formula: ...
0
votes
4answers
65 views

Testing the convergence of a series [closed]

Does the following series converge? $$\sum_{n=0}^{\infty}\frac{1}{n+3}$$ Please explain with any convergence test you used.
1
vote
1answer
58 views

How to analyze convergence or divergence of the integral $\int_1^{\infty}(t^2+\ln^2t)^{-1}dt$

Analyze convergence or divergence of the integral $\displaystyle\int_1^{\infty}(t^2+\ln^2t)^{-1}dt$ since $\displaystyle\int f(y)^{-1}dy=yf(y)^{-1}-F(f(y)^{-1})+C$ ...
19
votes
1answer
251 views

Does the series $\sum\frac{a_n}{a_{n+1}}\frac{1}{n}$ always diverge if $0<a_n<1$?

Does the series $$\sum_{n=1}^{\infty}\frac{a_n}{a_{n+1}}\frac{1}{n}$$ diverge for any $a_n$, satisfying $0<a_n<1$, $n=1, 2, 3\dots$ ?
1
vote
1answer
22 views

Proving convergence/divergence for $p$-series

I have an exam in Calc 2 coming up. As such, I am doing previous exams given by our current professor. However, the exams lack a solution set, so I will post the question, and the answer I wrote down ...
1
vote
1answer
40 views

Alternating Series Proof

I should be able to figure this out, but it has me a bit confused conceptually. I'm really just not sure how to approach it in a rigorous fashion. Any help? If $a_0, a_1, a_2, . . .$ is a decreasing ...
1
vote
3answers
57 views

Convergence test of a series? $\sum_{n=0}^\infty \frac{(-\pi)^n}{2^{2n+1}}$

What is the best way to test the convergence of the following series? My first guess is to use the Leibniz rule, but the exercise also asks to calculate it's sum, that makes me think that this is a ...
3
votes
0answers
34 views

Help proving(well, disproving) the convergence of $\sin^3(x)$

So I'm stuck on a question where it's asking for the power series and radius of convergence of $\sin^3(x)$ I've done the power series ok, but my problem is that when I apply the ratio test it's ...