0
votes
2answers
21 views

A question about limsup and limif

Could you please help me understand this question: Suppose $a_n$ is bounded sequence and $A<\liminf a_n$, $B>\limsup a_n$. Prove : $A<a_n<B$ for all n>N. It seems to me to simple to be ...
2
votes
1answer
32 views

Determine whether the series converge (adding fractions)

$$\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + ... $$ Help convert to summation. Not sure what test to use.
0
votes
1answer
20 views

Use comparison or limit comparison test to determine whether the series converge

Summation symbol $$\frac{(k^2+1)^{1/3}}{(k^3+2)^{1/2}} \ .$$
2
votes
2answers
49 views

How to prove convergence of $a_n$ if $(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$?

Could you give me some hint how to conclude convergence of $a_n$ from this feature : $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ From $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ we may conclude that ...
1
vote
1answer
29 views

Almost Sure convergence.

Given $(X_n, n\in\Bbb N)$ and $(Y_n, n\in\Bbb N)$ sequences of random Variables. For all $n\in\Bbb N$ it is : $X_n=Y_n$ almost sure. Now the question: Is then $P(X_n=Y_n \forall n\in\Bbb N)=1$? ...
0
votes
0answers
18 views

Consider the sequence $\{p_n\}_{n \in P}$… [closed]

Consider the sequence $S=\{p_n\}_{n \in P}$, where $p_n$ is the decimal expansion of $\sqrt{2}$ truncated at the $n$th decimal place (so $p_1=1.4, p_2=1.41, p_3=1.414$, etc.) If we work only in ...
1
vote
1answer
38 views

How to prove : If $a_{2n},a_{2n-1},a_{7n}$ converges than $a_n$ also converges.

Could you give me some hint how to prove this statement: If $a_{2n},a_{2n-1},a_{7n}$ converges than $a_n$ also converges. I think, obviously wrong, that if $a_{2n}\to a$ and $a_{2n-1}\to b$ than ...
1
vote
2answers
62 views

The limit of $\ln(n) - \ln(n^2 + 1)$ as $n\to\infty$

As $n\to\infty$, what is the limit of $\ln(n) - \ln(n^2 + 1)$ Using properties of logs and limits, I ended up with: $$ \ln \left(\lim \left(\frac{n}{n^2 + 1}\right)\right) $$ where lim is the ...
0
votes
1answer
24 views

Question about bounded sequence with two sub-sequential limits.

Could you please give me some hint how to deal with this question. Suppose $(a_n)$ is bounded sequence with 2 sub-sequential limits. Prove : there are real numbers A and B that ...
2
votes
2answers
81 views

If $\sum a_n$ converges then $\sum (-1)^n \frac {a_n}{1+a_n^2}$ converges?

Could you please give me some hint how to deal with this question: If $\sum a_n$ converges, does this necessarily mean that $\sum (-1)^n \frac {a_n}{1+a_n^2}$ must converge also ? Thanks.
0
votes
1answer
22 views

stats - limiting distribution of $X_i$

Suppose $P(X_{n}=i) = \frac{n+i}{3n+3},$ for $i = 0, 1, 2$. Find the limiting distribution of $X_{n}$. Is the cdf $F_{x_{n}}$ like: " if x = 0, $F_{x_{n}} = \frac{n}{3n+3}$; if x = 1, $F_{x_{n}} = ...
0
votes
2answers
19 views

stats - limiting distribution

Let $0 < p < 1$, and let $X_{n}$ have p.d.f. $f_{n}(x) = ( 1 – p ) ( n + 1 ) ( 1 – x )^{n} + p n x^{n – 1}$, for 0 < x < 1, zero elsewhere. Find the limiting distribution of $X_{n}$. ...
1
vote
2answers
41 views

Infinite series: which test

I'm having troubling deciding which test to use for this: $$\sum_{n=1}^\infty (-1)^n\arctan\left(\frac{\ln(n!)}{n+4^n}\right)$$ I tried altnerating test and ratio test but I couldn't get an answer. ...
1
vote
2answers
32 views

Sequence with an infinite amount of limit points

Find a sequence which has an infinite amount of limit points. I was thinking about using the bijective pairing function $\langle\cdot,\cdot\rangle:\Bbb N\times\Bbb N\to\Bbb N,\langle ...
3
votes
1answer
61 views

A Fibonacci series

Let $F_n$ be the $n^{th}$ term of the Fibonacci sequence. That is, $F_1 = F_2 = 1$ and $F_n$ is defined recursively for $n\geq3$ by $F_n = F_{n-2}+F_{n-1}$. It is a known fact that $$ ...
0
votes
0answers
39 views

Show convergence of an infinite sum by ratio test

I would like to show that $$\sum\limits_{n=1}^\infty \frac{1}{|n^x|}$$ converges. I was hoping to do this by the ratio test, since we haven't covered the integral test yet so we are not allowed to use ...
0
votes
0answers
46 views

When the series $\sum\limits_{n=1}^{\infty} (\sqrt{n+a}-\sqrt[4]{n^2+n+b})$ converge and diverge

How to, depending on the real parameters a and b, determine when will the series converge and when will it diverge: $\sum\limits_{n=1}^{\infty} (\sqrt{n+a}-\sqrt[4]{n^2+n+b})$. I got this for homework ...
5
votes
4answers
265 views

Convergent or divergent

For homework (Calculus 2) I have to determine does this series converge or diverge and I don't know how to start: $$\sum\limits_{n=1}^{\infty} \dfrac {\ln(1+e^{-n})}{n}. $$
2
votes
2answers
98 views

Does the series $\sum\limits_{n=1}^{\infty}n\tan\left(\frac {\pi}{2^{n+1}}\right)$ converge or diverge?

Does the series $\sum\limits_{n=1}^{ \infty}n\tan\left( \dfrac { \pi}{2^{n+1}}\right)$ converge or diverge? My idea was to use the limit comparison test and $\sum\limits_{n=1}^{\infty} \dfrac ...
0
votes
1answer
39 views

Convergence and Divergence [duplicate]

Suppose that the series $\sum_{n=1}^\infty a_n$ is conditionally convergent. Prove that the series $\sum_{n=1}^\infty n^2a_n$ is divergent. How should I start to prove this? I have absolutely no ...
-1
votes
1answer
93 views

Does the series converge or diverge as n-> inf: cos(n)/n^3

It's pretty obvious that it converges, seeing as n^3 continues getting larger, and cos(n) is bounded by 1 and -1. The ratio and root tests are useless. I was just wondering if I could use the ...
1
vote
1answer
30 views

Determine for which nonnegative real number $\displaystyle\alpha$ the series converges

Determine (with proof) for which nonnegative real numbers $\alpha$ the series $$ \displaystyle\sum_{n=1}^{\infty}\left(\frac{1}{n^\alpha}-sin\left(\frac{1}{n^\alpha}\right)\right) $$ OK, first I ...
1
vote
2answers
56 views

How do I calculate the Radius of convergence of this sum [closed]

What is the radius of convergence? $$\sum_{n=0}^{\infty} n^3 (5x+10)^n$$
2
votes
1answer
66 views

How can we show that the following sequence converges?

$(a_n)$ is a bounded sequence with the following condition $a_{n+1}\geq a_n-\frac{1}{2^n}$ The sequence converges, but how do we show it?
1
vote
2answers
67 views

Prove that the series converges

Let $(a_n)$ be a bounded positive and monotone increasing sequence. I need to show that $\sum (1-\frac{a_n}{a_{n+1}})$ converges. My approach was as follows: Let $B=sup (a_n)$ Then since the ...
1
vote
0answers
52 views

Uniform Convergence of $f_n(x)=1+\cos(x/n)$

How to see that the series of functions $f_n(x)=1+\cos(x/n)$ does not converge uniformly on $\mathbb{R}$. To prove $f_n(x)$ does not converge uniformly to $1$ is easy. Let $x=0$, then $|f_n(0)-1| = 1 ...
2
votes
1answer
30 views

Show an iterative fixed point method does not converge

I was asked the following question: let $g(x)=\frac{30}{1+x}$. Notice that $g(5)=5$. Is there an $\epsilon >0$ such that the series $\{x_k\}_{k=0}^{\infty}$ defined by $x_{k+1}=g(x_k)$ and ...
0
votes
0answers
39 views

Show an iterative method converges and find the rate of convergence - Numerical Analysis

We are defining an algorithm as follows: Let $f(x)$ be a function with a root in $[a,b]$. We define a series $\{x_k\}_{k=1}^{\infty}$ as follows: $x_{k+1}=x_k-f(x_k)\frac{b-a}{f(b)-f(a)}$. Does ...
0
votes
4answers
149 views

Ratio test: $n^\sqrt{n}$

I need to determine the radius of convergence of: $$\sum_{n=1}^\infty z^n n^\sqrt{n}$$ I have, by use of the ratio test, written: (Because I know it tends to 1) $$\lim_{n\to\infty} ...
0
votes
1answer
24 views

Sequence and subsequence convergence

Let {$x_n$} be a sequence and $x \in \mathbb{R}$. Prove {$x_n$} converges to $x$ if and only if for every subsequence {$x_{n_k}$} of {$x_n$} there exists a subsequence {$x_{n_{k_j}}$} of {$x_{n_k}$} ...
1
vote
1answer
48 views

Proving if there exist a subsequence converging to $1/n$, there exist a subsequence converging to $0$.

Let {$x_k$} be a sequence and suppose for every $n \in \mathbb{N}$ there exist a subsequence converging to $1/n$. Prove there exist a subsequence converging to $0$. Is this as simple as saying since ...
1
vote
3answers
29 views

Proving a sequence converges by defining the sequence recursively.

Let $a > 0$. Choose $x_1 >$ $\sqrt a$. Define a sequence {$x_n$} recursively as $x_{n+1} = 1/2(x_n + a/x_n)$ for $n > 1$. Prove that lim$x_n = \sqrt a$. I think I first want to prove that ...
0
votes
1answer
43 views

If $\sum a_n$ converges and for almost all n $a_n>0$, does it mean that the series $\sum a_n$ converges absolutely?

If $\sum a_n$ converges and for almost all n $a_n>0$, does it mean that the series $\sum a_n$ converges absolutely ? Thanks.
0
votes
1answer
28 views

Numerical Analysis - show something about the rate of convergence

We are given an iterative method for finding roots, $x_{n+1}=g(x_n)$, we are given the rate of convergence of this method is $p$, and also that: $$\lim _{n \to \infty} \frac{|e_{n+1}|}{|e_{n}|^p} = ...
1
vote
2answers
30 views

Sequence convergence to zero

Suppose that {$x_n$} is a sequence of positive numbers and $\hspace{70mm}$ lim ($\frac {x_{n+1}}{x_n}$) = $L$ Show that if $ L < 1$, then the lim $x_n = 0$ We know that if $L < 1$ then ...
0
votes
1answer
37 views

Convergence of Real Number sequences

(a) Suppose {$a_n$} $ \rightarrow a$. Let {$b_n$} be the sequence defined by: $\hspace{70mm} $$b_n = \frac{a_1+...+a_n}{n}$ Show that {$b_n$} $\rightarrow$ $a$ (b) Let $a_n$ = {$(-1^n)$}. Show ...
0
votes
1answer
85 views

Proof of convergence of an infinite product

a) Show that $\Pi_{n=1}^\infty x_n$ converges if and only if for all $\varepsilon>0$ there exists an $N$ such that for all $m\ge n\ge N$, $\left|x_nx_{n+1}\cdots ...
0
votes
1answer
107 views

Prove -n^2 diverges to negative infinity

Prove directly that the following sequence diverges to negative infinity $a_n = -n^2$ I understand that the sequence will diverge to negative infinity. I know that I must somehow integrate an $n$ ...
5
votes
1answer
144 views

Two sequences are subsequences of one another, one converges. Are they the same?

Let $a_n$ converge to a. Let $b_n$ be a subsequence of $a_n$ and $a_n$ be a subsequence of $b_n$. Are they the same? I've tried showing they are by contradiction, the Bolzano-Weierstrass theorem ...
0
votes
1answer
41 views

Does the series $\sum_{n=1}^\infty \frac {x^n} {a^n-b^n}$ converge?

Suppose there are the series $\displaystyle\sum_{n=1}^\infty \dfrac {x^n} {a^n-b^n}$, $\,a>b>0$. I tried to calculate radius of convergence by $\dfrac1R=\limsup\sqrt[n]{\dfrac 1 {a^n-b^n}}$ but ...
2
votes
0answers
75 views

Pointwise convergence on a complete metric space

Let $ f_n :X \rightarrow R $ be sequence of continuous function on complete metric space $(X,d)$, which convergence pointwisely to $ f: X\rightarrow R $ mean that $ f_n(x)\rightarrow f(x)$ $ ...
2
votes
0answers
71 views

Discontinuous function on Q

let $ f_n :X \rightarrow R $ be sequence of continuous function on complete metric space $(X,d)$, which convergence pointwisely to $ f: X\rightarrow R $ mean that $ f_n(x)\rightarrow f(x)$ $ ...
1
vote
1answer
31 views

Show that $A$ is non-compact

I have a problem: For $C\left [ 0,1 \right ]=\left \{ x:\left [ 0,1 \right ] \to \Bbb R \ \text{is continuous on } \left [ 0,1 \right ] \right \}$, with a norm: $$\left \| x \right \|=\sup_{t\in ...
1
vote
1answer
64 views

Limit of function with three variables

given is $f(x,y,z) = \frac{xy + yz}{x^2+y^2+z^2}$. I want the limit for this function for $(x,y,z)\to(0,0,0)$. Since I haven't done this ($=$calculating a limit for a function with more than $1$ ...
1
vote
1answer
66 views

Function Series pointwise/uniform/compact convergence

In the following excercise I have to tell whether the function-series is a) pointwise convergent b) uniform convergent and/or c) compact convergent on intervall $[a, \infty]$ (a > 0). The series is: ...
1
vote
0answers
25 views

Testing for convergence using comparision theorem

I know that If $f(x)\geq g(x)\geq 0 \forall x$ on the interval $[a,\infty)$ then, if $f(x)$ is convergent,$g(x)$ also must converge. In doing a exercise, I have come across a problem where the use ...
2
votes
1answer
78 views

That sequence is convergent pointwise but not uniformly convergent, but for what $\varepsilon$?

I'm trying for too long time solve this but without success. I'm tired and need help. Show that sequence of funcions converges pointwise but not uniformly convergent: $$ ...
3
votes
2answers
123 views

Does the series $\sum_{n\ge1}\left(-1\right)^n\frac{\cos\left(\alpha n\right)} {\sqrt n}$ converge?

Does the series $\sum_{n\ge1}\left(-1\right)^n\frac{\cos\left(\alpha n\right)} {\sqrt n}$ converge ? I tried to deal with this problem this way. Let $S_k$ be a sequence of partial sums of the given ...
0
votes
1answer
46 views

If the series $\sum_{n\ge 1}f_n \left(x \right)$ converges but not uniformly does the sequence $\left(f_n\left(x\right)\right)$ converge?

From Cauchy Criterion for Uniform Convergence we can conclude that if the series $\sum_{n\ge 1}f_n \left(x \right)$ converges uniformly than the sequence $\left(f_n\left(x\right)\right)$ converges ...
1
vote
1answer
74 views

Could non-continuous sequence of functions converge uniformly to continuous function?

If continuous sequence $ \left( f_n\left(x\right) \right)$ converges uniformly to function $f\left(x\right)$ in some interval of real numbers, than $f\left(x\right)$ must be also continuous. But if ...