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

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2
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1answer
26 views

How to chose the resembling series in the comparison test

I am having some difficulties choosing the resembling series for the comparison test. This problem is an example: $$ a_n=\sum_{n=1}^\infty \left(\frac{5}{n}-\frac{5}{n^2}\right)$$ The way I think ...
0
votes
1answer
41 views

Formula for Taylor Polynomials

I'm studying from Calculus: Early Transcendentals, by Briggs & Cochran, and the authors give the definition of Taylor Polynomials as: $p_n(x) = \sum \limits_{k=0}^n c_k(x-a)^k$, where the ...
1
vote
3answers
81 views

Converging Sequences

How can I show that the sequence $\{x_n\}_{n=1}^{\infty}$ defined by $$x_{n+1} = \frac{x_n(x_{n}^2 + 3a)}{3x_{n}^2 + a}$$ is convergent? I don't think plugging this into the convergent-sequences ...
1
vote
3answers
56 views

Method to find sum of a series

I am trying to find the sum of the series, $\large\Sigma_{n=0}^\infty \large\frac{(-1)^n}{5^n}$ and I have no idea even how to start. The only way I know to find sums is: 1)By geometric series ...
0
votes
1answer
16 views

Approximating the value of the limit of a sequence defined recurrently

Suppose I have a sequence defined by recurrence, i.e. $x_{n+1}=f(x_n)$ for some $f:\mathbb{R}\to \mathbb{R}$, and $x_0\in \mathbb{R}$. Suppose that $f$ is $K$-Lipschitz for some $K<1$. Then $f$ has ...
3
votes
0answers
86 views

Functional inequalities involving cubing and incrementing

Consider the set $S$ of positive increasing invertible functions $f$ satisfying: $$f((x+1)^3-1)≤(f(x)+1)^3-1$$ $$f(x^3)≥(f(x))^³$$ $$f(x)+1≤f(x+1)$$ for all positive real $x$. Clearly the identity ...
0
votes
1answer
33 views

Sandwich Theorem for Sequences

I am to determine if the sequence converge or diverge, and find the limit in case of convergence. $a_n=\frac{(-1)^{n+1}}{3n-4}$ The solution manual states that I can apply the Sandwich Theorem, ...
8
votes
1answer
205 views

Show the sequence converges to M

Assume $f : [a,b] \to R$ is continuous and $f(x) \ge 0$ for all $x \in [a,b]$, and $M = \sup\{f(x) : x \in [a,b]\}$. Show that $$\lim_{n\to\infty}\left[\int_a^bf(x)^ndx\right]^{1/n}$$ converges to ...
1
vote
2answers
29 views

$x_n\le y_n\le x_{n+2}$

Could any one tell me which of this is correct statement?I did not get any hint. I get $x_1\le y_1\le x_3$ $x_2\le y_2\le x_4$ etc but how to know all these $4$ properties?
2
votes
2answers
62 views

Estimating integral $\int_0^{0.5} \ln(1+\frac{x^2}{4})$

Estimate the definite integral $\int_0^{0.5} \ln(1+\frac{x^2}{4})$ with an error of at most $10^{-4}$, using the Alternating Series Estimation Theorem. My approach is as follows: I found the ...
0
votes
1answer
30 views

Can someone please show the steps from the question to the answer?

I have a question which has an answer without steps in between. Can someone help? $$m(t)=E\left(e^{tY}\right) = \sum_{y=0}^n \binom{n}{y}\left(pe^t\right)^yq^{n-y} = \left(pe^t+q\right)^n$$ To not ...
0
votes
2answers
46 views

If $\sum_{k = 1}^{\infty} a_{k}$ diverges, then one of its rearrangement converges to 1

If $\sum_{k = 1}^{\infty} a_{k}$ diverges, then there exists a rearrangment of $\sum_{k = 1}^{\infty} a_{k}$ that converges to 1. This is not true. A counter example I came up with is $\sum_{k = ...
1
vote
2answers
30 views

Posivite Values P For Which a Series Converges

Consider the following series $$ \sum_{n=1}^{\infty}\frac{(n!)^2}{(Pn)!} $$ For which positive values of $P$ does the series converges? Not sure if the Ratio Comparison Test would be useful here: ...
2
votes
1answer
56 views

Closed-form of prime zeta values

The prime zeta function is defined as $$P(s)=\sum_{p\,\in\mathrm{\mathcal P}} \frac{1}{p^s},$$ where $\mathcal P$ is the set of prime numbers. It converges for all $\Re(s)>1$. There is a related ...
-1
votes
2answers
51 views

Evaluation of $\lim_{n\to\infty} \frac{1}{\ln n}\sum_{k=1}^n \frac{k}{k^2+1}$

I want to evaluate $$ \lim_{n\to\infty} \frac{1}{\ln n}\sum_{k=1}^n \frac{k}{k^2+1} $$ I can already see that $$\lim_{n\to\infty}\frac{1}{\ln n} = 0$$ so how do we go about solving this?
1
vote
2answers
29 views

Is this claim true? (About the sequence $\{a_n^{\frac{1}{n}}\}$)

Suppose that $\{(a_n)^{\frac{1}{n}}\}\subseteq \mathbb{R}$ converges and denote with $\alpha$ its limit ($a_n>0$ for all $n$). Is it true that $\{(a_{n+1})^{\frac{1}{n}}\}$ converges to the same ...
0
votes
1answer
50 views

Proof of convergence and determination of the limit for a series, with a nested sum.

I happend to proof quite a few series convergent, however, this one is driving me nuts: My Task: Show that the following series converges, and determine its limit. $$\sum_{\nu = ...
1
vote
1answer
104 views

Numerical value of $\sum_{p \in \mathcal P} \frac1{p\ln p}$

In this question we determine that the series $\sum_{p \in \mathcal P} \frac1{p\ln p}$ converges, where the sum runs over primes. As I see the convergence is really slow. The partial sums for given ...
0
votes
1answer
119 views

Find the values of $a$ and $b$ such that $b = \lim_{n \to + \infty } n^a \sum_{k = 1}^n {\frac{1}{{\sqrt k }}}$

Evaluate $a ;b$ values which satisfy this equality : $$b = \lim_{n \to + \infty } n^a \sum_{k = 1}^n {\frac{1}{{\sqrt k }}}$$ My solution $$ \sqrt n \le \sum\limits_{k = 1}^n ...
11
votes
3answers
504 views

$\sum_{p \in \mathcal P} \frac1{p\ln p}$ converges or diverges?

We will denote the set of prime numbers with $\mathcal P$. We know that the sum $\sum_{n=1}^{\infty}\frac1n$ and $\sum_{n=2}^{\infty}\frac1{n\ln n}$ diverges. It is also known that $\sum_{p \in ...
4
votes
2answers
97 views

Hard limit: $\lim_{n \to +\infty} \left(\sin\left(2\pi(k!)x\right)\right)^n$

I've stumbled on a particularly hard limit problem: Evaluate the following limit: $$\lim_{n \to +\infty} \left(\sin\left(2\pi(k!)x\right)\right)^n\text{, with $n, k \in \mathbb{N}$ and $x \in ...
1
vote
1answer
40 views

Convergence implied by partial sequences. [duplicate]

I am not sure how to deal with this Question: Show, that the sequence $(a_{n})_{n \in \mathbb{N}}$ converges towards the limit $a \in \mathbb{R}$, exactly when every subsequence $(a_{n_{k}})_{k \in ...
1
vote
1answer
42 views

Does this sequence of integer products have a name?

Suppose that I have a product of, say, $n=4$ integers starting with one and ending with four $1234=4!=24$. Now I construct all products of four positive integers $1,2,3$ and $4$ with repetition such ...
2
votes
0answers
109 views

Can Wynn's $\epsilon$ algorithm be used for sequence limit?

Let's assume we have a sequence $(a_n)$, which converges to some limit $L = \lim_{n\to\infty} a_n$. However, we are able to calculate only first $N$ terms of the sequence. It is clear that, in ...
2
votes
2answers
122 views

Example 3.53 in Baby Rudin

Consider the convergent series $$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \ldots$$ and one of its rearrangements $$1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + ...
1
vote
1answer
33 views

Finding the limit of a series [duplicate]

I could just prove that the sequence is bounded but couldn't find the exact limit.
4
votes
3answers
110 views

Prove that $\sum_{n=0}^{\infty }\frac{(2n+1)!!}{(n+1)!}2^{-(2n+4)}=\frac{3-2\sqrt{2}}{4}$

How can I prove that $$\sum_{n=0}^{\infty }\frac{(2n+1)!!}{(n+1)!}2^{-(2n+4)}=\frac{3-2\sqrt{2}}{4}$$
1
vote
1answer
150 views

Show that the following definitions all give norms on $S_F$

$S_F$ is the space of all real sequences $\mathbf a=(a_n)_{n=1}^{\infty}$ such that each sequence $\mathbf a\in S_F$ is eventually zero. Show that the following definitions all give norms on $S_F$, ...
1
vote
2answers
37 views

Finding the convergence/divergence of a series

In the series: $C_n = \dfrac{9^n}{n!}$ How do you figure out whether it converges or diverges? I've tried writing it out a little bit, but it doesn't really help much . . .
5
votes
1answer
69 views

$\sum_{i=1}^{\infty}a_n*b_n $ converges for all $\lim_{n \rightarrow \infty}b_n = 1$, show that $a$ converges absolutely

So this is given: $\sum_{n=1}^{\infty}a_n*b_n $ converges for all sequences $(b_n)$, such that $\lim_{n \rightarrow \infty}b_n = 1$. Somehow it should be showable that $(a_n)\,$converges absolutely, ...
0
votes
1answer
31 views

Is the following function closed under addition?

Consider the set of all functions on [0,1] of the form $h(x) = \Sigma_{j=1}^na_je^{b_jx},$ where $a_j, b_j \in \mathbb{R}$. Is this set closed under addition? That is consider $f(x), g(x) \in ...
1
vote
2answers
481 views

Is this series convergent or divergent? (with factorials/alternating series)

Determine the series: $$\sum_{n=1}^{\infty} {\sqrt{\frac{n!}{(n+2)!}}}$$ This wouldn't be a alternating series since there is no $(-1)^n$. And I don't think taking a risk of using The Ratio Test ...
1
vote
2answers
49 views

Determine whether this series is convergent or divergent. (sins in series)

Determine where the series: $$\sum_{n = 1}^\infty \frac{\sin^4(n)}{n^2+1}$$ Would the comparison test not work? Since there is a $\sin^4$ I would not be able to take its dx easily, would there be a ...
1
vote
3answers
46 views

Does this series converge or diverge? (Limit Comparison Test)

Determine (with justification) whether the series: $$\sum_{n = 2}^{\infty} \frac 1{n\sqrt{(n^2-1)}}$$ Since this will be positive at all times I thought it was a good candiate for the Limit ...
1
vote
4answers
57 views

The integral diverges

Prove that $\displaystyle \int_{0}^{\pi}\left \lfloor \cot x \right \rfloor\,dx$ diverges. Proof:(or some part of it anyway) $$\begin{aligned} \int_{0}^{\pi} \left \lfloor \cot x \right ...
1
vote
1answer
114 views

Comparison between lebesgue integral and riemann integral of $f(x)=x^2$ in $[0,2]$

If we have an example $f(x)=x^2$ let's say for $[0,2]$. In lebesgue integral, I already use a sequence of function $f_n(x)$ as approximation to $f(x)$ ($f_n(x)$ converges to $f(x)$) which is stated ...
0
votes
3answers
37 views

Infinite Series Convergence using Comparison

So we know that the sum $$ \sum_{n= 1}^{\infty}\dfrac{n}{(n^2 + 1)}$$ is divergent because it is practically the same function as $\dfrac{1}{n}$ which is also divergent. But what official test ...
0
votes
1answer
19 views

Values of x for the the geometric series converges

Problem is, $4^n (x+2)^n$ with $n$ going to $0$. After applying the ratio test I'm getting the absolute value of $4(x+2)$. If the $4$ wasn't there I'm pretty sure the interval would be $(-2,2)$, but ...
1
vote
2answers
58 views

Evaluating $\sum 1/(2n+1)!$

I am not sure how to evaluate the series. Apparently it equals the hyperbolic sin at 1, but I don't see it. Not sure if I need to start breaking down the factorial or how to even begin the problem. ...
1
vote
1answer
30 views

Need answer for this series

$\lim_{n\rightarrow\infty}\frac{1}{2^n\sqrt{2^n}}\sum_{k=1}^{2^{n+1}}(k-1)(\sqrt{k}-\sqrt{k-1})$ I stuck on the series. Please help. Thanks
4
votes
1answer
275 views

Problem with infinitely many converging subsequences

Knowing that $ \forall $ $k \in \mathbb N, k>=2$, the subsequence $(a_{kn})$ converges, does that mean that $(a_n)$ converges? I suppose this seems true, but I'm having problems sketching a ...
2
votes
0answers
51 views

Find a closed formula for this sequence

Let $A(n, m)$ be functions on $\mathbb{N}^2$ such that $A(n, 0)=n$ and $A(m, n)=A(n, m)$ and that $$ A(m, n)=\frac{mA(m-1, n)+n A(m, n-1)}{m+n}. $$ Question: Is there a closed formula for $A(n, n)$? ...
2
votes
1answer
29 views

Find infinite set for which the series diverges

I'm looking to clarify the meaning of a question, and would greatly appreciate any feedback. Given a function $f_n(x)$, I am to construct an infinite set S such the series ...
1
vote
3answers
49 views

Sum of $\sum_{n=0}^{\infty }\frac{1}{4^{(n/3)+1}}$

Find the sum of $$\sum_{n=0}^{\infty }\frac{1}{4^{(n/3)+1}}$$
1
vote
1answer
307 views

Prove that if A_n converges to λ than square root of A_n converges to square root λ?

Let an be a sequence of non-negative numbers converging to some number λ > 0. Prove that √an converges to √λ. (Hint: Note that multiplying and dividing dist(√an,√λ) by (√an + √λ) yields after a ...
2
votes
1answer
82 views

Is it always true that no closed forms exists for any divergent series?

Having seen many questions regarding finding closed form of integrals or infinite series, and some users providing either the final answer or detailed solution, and also reading how one finds a closed ...
0
votes
1answer
32 views

Find a sequence that is divergent but its average converges.

Let ${t_n}=\frac{s_1+s_2+...+s_n}{n}\text{ where } n \ge 1$. Find an example showing ${t_n}$ may converge even though ${s_n}$ diverges. Can anyone think of an example?
0
votes
2answers
52 views

Convergence and limited sequence

I am given two sequences: $\{a_n\}$ which converges towards 0 and $\{b_n\}$ which is a bounded sequence. I have to show that $\{b_na_n\}$ is converging towards 0. I seem to be quite stuck on how to ...
1
vote
1answer
84 views

series and inequality

I found this homework in an old paper written Let $n\in \mathbb{N}^*$ and $x_1,x_2,\ldots,x_n \in \mathbb{R}$ such that $ \sum_{i=1}^{n}\left|x_{i}\right|=1$ and $\sum_{i=1}^{n}x_{i}=0$ ...
3
votes
3answers
97 views

Power series expansion involving non integer exponent

I'm working on a real and complex analysis course right now and one power series question has me really stumped: I'm not sure what to do with the non integer in the exponent, as my initial plan of ...