Tagged Questions
1
vote
4answers
36 views
Is $(a_n)_n$ with $|a_{n+2}-a_{n+1}| ≤ q|a_{n+1}-a_n|$ a Cauchy sequence?
Let $0 < q < 1$ and $(a_n)_n$ be a squence with $|a_{n+2}-a_{n+1}| ≤ q|a_{n+1}-a_n|$ for all $n ∈ ℕ$.
I need to show that this is a Cauchy sequence. I'm not sure how to start this one, as we ...
0
votes
1answer
31 views
Show convergence for this sequence only by using the definition
I need to prove convergence for
$(b_n)_{n ∈ ℕ}=\left(\frac{(-1)^nn}{2n+1}\right)_{n∈ℕ}$ and also show the limit.
I may only use the following definition: $∀ɛ > 0∃n_0∈ℕ∀n≥n_0:|a_n-a|< ɛ$.
So far ...
5
votes
2answers
48 views
“Nearly” Harmonic Series
It's well known that
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1+\varepsilon}} < \infty, \ \forall \varepsilon >0.
$$
What happens if we replace $\varepsilon$ with $\varepsilon_n \downarrow 0$?
...
1
vote
1answer
20 views
Equicontinuity and uniform convergence 2
Let $\{f_n\}_n$ be a sequence of real valued functions on a compact metric space $K$. Suppose that for all $x$ we have $f_n(x) \to f(x)$ as $n \to \infty$ and that the family $\{f_n\}_n$ is ...
1
vote
1answer
49 views
Conditions for taking a limit into an infinite sum
Suppose $f\left(x\right)={\displaystyle \sum_{n=0}^{\infty}g_{n}\left(x\right)}$ under what conditions is it true that: $$\lim_{x\to c}f\left(x\right)={\displaystyle \sum_{n=0}^{\infty}\lim_{x\to ...
2
votes
1answer
37 views
Radius of convergence of a power series with Bernoulli numbers
Say, we use the definition:
Bernoulli numbers arise in Taylor series in the expansion $$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k \frac{x^k}{k!}$$
and then derive power series representations of the ...
1
vote
1answer
25 views
Approximating Lipschitz funtion by $C^1$ function.
Let $G:\mathbb{R}\to\mathbb{R}$ be a Lipschitz function and $L$ its Lipschitz constant. Suppose that $G(0)=0$. It is known that $G$ is almost everywhere differentiable and $G'\in L^\infty$ with ...
9
votes
4answers
82 views
$a_{n+f(n)}-a_{n}\rightarrow0$ implies convergence?
$(a_{n})$ is a sequence of reals. Say $a_{n+f(n)}-a_{n}$ tends to 0 as n tends to infinity for every function f from the positive integers to the positive integers. Does this imply that $a_{n}$ is ...
0
votes
1answer
28 views
Did there exists a counterexample says that the Dominated convergence Theorem must has a control function rather than the limits function?
Recall the Dominated Convergence Theorem is:
Let $\{f_n\}$ be a sequence of real-valued Lebesgue measurable functions on some open (in fact measurable will sufficient) subset of $E\subset R^d$. ...
0
votes
2answers
26 views
convergence question
Let $p > 0$. Prove that $\lim_{n \to \infty } (n^{p})^{1/n} = 1$
You may use that the function $f(x) = x^{p}$ is continuous at $x = 1$.
I know that I have to show that $ \left ...
1
vote
2answers
39 views
Sequence convergence of positive numbers
Suppose that $\{a_j\}_j$ is a sequence of real numbers. Suppose for all $j$, $a_j \geq 0$ and the sequence $b_j = \frac{a_j}{1 + a_j}$ converges to $0$.
I wish to prove that $a_j$ converges to $0$.
...
2
votes
2answers
63 views
A question on the convergence of a Taylor series of some prominent function
The function $f:\mathbb{R}\to\mathbb{R}$ defined by
$$
f(x)=\begin{cases}e^{-\frac{1}{x^2}} &if &x\neq 0\\
0 & else \end{cases}
$$
is a prominent example of a function whose Taylor series ...
0
votes
2answers
53 views
Monotonic sequence with a bounded subsequence
What is the proof of the following theorem?
"If a sequence is monotonic and has a bounded subsequence, then it is also bounded."
3
votes
1answer
43 views
Estimate of a summation
Show that for each $\alpha \in (0,1)$ there exists a constant $C_\alpha$ such that
$$
|F_\alpha(x)| \leq C_\alpha |x|^\alpha
$$
for all $x \in \mathbf{R}$ where $F_\alpha$ is given as
$$
...
0
votes
2answers
47 views
Ratio test and the radius of convergence
Let
$$
\sum_{n=0}^\infty c_n (z-a)^n
$$
be a power series. If the value
$$
r=\underset{n\to\infty}{\lim}\left|\frac{c_n}{c_{n+1}}\right|
$$
exists (the limit exists and is a real number), it is the ...
2
votes
0answers
64 views
When $\ell^2$-convergence implies $\ell^1$- convergence?
Consider a sequence $(x_n)_{n\in\mathbb N}$ in $\ell^1$ (sequences taking their values in $\mathbb R$), where $x_n=(x_{i,n})_{i\in\mathbb N}$.
What are sufficient conditions on the sequence ...
1
vote
2answers
37 views
Prove that a sequence diverges
Let $b > 1$. Prove that the sequence $\frac{b^n}{n}$ diverges to $\infty$
I know that I need to show that $\dfrac{b^n}{n} \geq M $, possibly by solving for $n$, but I am not sure how.
If I ...
0
votes
1answer
29 views
Study the convergence of this sequence of functions
I have the following sequence of function:
$$f_n(\lambda)=\bigg[\alpha-i\bigg(\lambda+\frac{1}{n}\bigg)\bigg]^{-1}-\bigg[\alpha-i\lambda\bigg]^{-1},\,\,\,\alpha\neq 0$$
and I have to study its ...
1
vote
2answers
33 views
An example of uniform convergence on compact sets but not uniform convergence?
As the title suggests, I want to find an example where a sequence of continuous functions $\{f_n\}$ converges uniformly on compact sets to a continuous function $f$, and yet the convergence is not ...
0
votes
1answer
48 views
How to prove a telescoping series converges ???
Let ${a_k}$ be a real sequence, such that $a_k \rightarrow 3, $as k$ $$\rightarrow \infty$ .
Prove that $\sum_{k=1}^\infty (a_k- a_{k+3}) = a_1 + a_2 +a_3 - 9$.
I know this is a telescoping series ...
1
vote
2answers
89 views
Continuous functions uniformly convergent to a function, metric spaces, equivalent conditions
Let $X, \ (Y, d)$ be metric spaces, $f_1, f_2, \ldots \ : X \rightarrow Y$ be continuous functions, $f: X \rightarrow Y$ an arbitrary function. Prove that the following condtions are equivalent:
1) ...
0
votes
2answers
26 views
Limit function of pointwise convergence is always bounded?
If $\{f_n \colon [a,b] \rightarrow \mathbb R\}$ is a sequence of bounded functions converging pointwise to $f \colon [a,b] \rightarrow \mathbb R$, then $f$ is bounded.
Is the statement above ...
1
vote
2answers
36 views
Convergence of $\max_{0\le i\le n}|f(i/n)|$
Suppose that $f\colon [0,1]\to\mathbb R$ is a continuous function. How can I prove that
$$\max_{0\le i\le n}\biggl|f\Bigl(\frac in\Bigr)\biggr|\to\sup_{0\le x\le1}|f(x)|$$
as $n\to\infty$?
Any help ...
1
vote
1answer
40 views
Convergence of $x_n=\sqrt{2(x_{n-1}-\log(1+x_{n-1}))}$
$x_n=\sqrt{2(x_{n-1}-\log(1+x_{n-1}))}$ with $x_0=a\ge0$
I calculated some values for $x_n$ with $a=0,1,2$ and recognized its monotonically decreasing (It converges to $0$). Now the standard approach ...
0
votes
2answers
39 views
Are these converging or diverging
I am having trouble working out the convergence of these series and was wondering if I could please have some assistance
a) $\displaystyle\sum_{n=0}^\infty\sin(e^n)\frac{n}{n^3+1}$
and
b) ...
1
vote
4answers
79 views
check the convergence of the integral $\int_{0}^{\infty}\frac{1}{x\log x}\,dx$
Help me on checking the convergence of the integral $$\int_{0}^{\infty}\frac{1}{x\log x}\,dx$$
I have tried it in this way $$\int_{0}^{\infty}\frac{1}{x\log x}\,dx=\int_{0}^{\frac{1}{2}}\frac{1}{x\log ...
2
votes
2answers
71 views
Does the series always diverge?
Investigating the behavior of the following series:
$$\sum_{k=2}^\infty \frac{1}{\log^{p}k}$$
I broke it into 3 parts:
If $p = 0$ then it's just an infinite summation of ones, which diverges
If $p ...
5
votes
2answers
67 views
Does $f_n \to 0$, a.e., implies $\int_{\mathbb R} \sin(f_n(x)) dx \to 0$, when each $f_n \in L^1$
Let $\{f_n\}$ be a sequence of $L^1(\mathbb R)$ functions converging a.e. to zero.
Does
$$
\lim_{n\to \infty} \int_{\mathbb R} \sin(f_n(x)) dx = 0?
$$
I think the answer is no, but I can't find a ...
0
votes
2answers
70 views
check the convergence of the improper integral$\int_{0}^{1}\frac{x^{p-1}+x^{-p}}{1+x}\,dx$
How to check the convergence of the improper integral$$\int_{0}^{1}\frac{x^{p-1}+x^{-p}}{1+x}\,dx$$
I can only check that the integral is divergent for $p\geq1$, help for the cases when $p<1$.
...
3
votes
5answers
120 views
Sequence of continuous functions which converges to a continuous limit
Any help with this: construct a sequence of continuous functions defined on $ [0,1] $ which converges pointwise but not uniformly to a continuous limit ?
Thank you.
0
votes
3answers
41 views
finding values for absolute convergence
Find all values of real number p or which the series converges:
$$\sum \limits_{k=2}^{\infty} \frac{1}{\sqrt{k} (k^{p} - 1)}$$
I tried using the root test and the ratio test, but I got stuck on ...
2
votes
2answers
25 views
Radius of Convergence for $\sum \frac{[1\cdot 3 \cdots (2n-1)]^2}{2^{2n}(2n)!}x^n$
I'm trying to find the radius of convergence for this series:
$$\sum \frac{[1\cdot 3 \cdots (2n-1)]^2}{2^{2n}(2n)!}x^n$$
so I have,
$$R=\lim_{n\to\infty} \frac{[1\cdot 3 \cdots ...
0
votes
2answers
49 views
Composition of a continuous function with functions that converge uniformly
Here is problem that appeared in one of the past final exams for my introductory real analysis course, that I am having hard time to solve. It is Question 5 in 8 of the following file:
...
2
votes
2answers
43 views
Absolute Convergence of a Series
Find all values of real number p for which the series converges absolutely
$$\sum_{k=2}^{\infty} \frac{1}{k\, (\log{k})^p}$$
2
votes
1answer
38 views
Convergence and monotonicity of roots of $\frac{1}{1 + x - \exp(-x)} = n$
We suppose that $\forall x\in \mathbb{R}\setminus{0} \quad f(x)=\dfrac{1}{1+x-\exp(-x)}$ , and $x_n \in ]0;+\infty[$ as $x_n$ is an unique solution of the equation $f(x)=n$ on $]0;+\infty[$.
How can ...
1
vote
2answers
58 views
Show $\sum \frac{3^{2k+1}}{k^{2k}}$ converges
I'm pretty certain it can be shown using the ratio test; I simplified $a_k+1/a_k$ to $[ 9(k)^{2k} )/( (k+1)^{2k+2} ]$ then let lim k->inf a_k+1/a_k = x thus lnx = lim k->inf ln[ 9(k)^(2k) )/( ...
0
votes
1answer
49 views
Prove convergence of improper integral using change of variable.
This may be trivial, but I could use some help...
Consider a real function $f: (0,1) \rightarrow \mathbb{R}$, continuous, positive, but not necessarily bounded. Let $g: [0,1] \rightarrow [0,1]$ be a ...
0
votes
1answer
34 views
Prove rearrangement of harmonic series tends to 1 or -1
Prove that a rearrangement of the series $\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n}$ has a subsequence that tends to 1 or -1 or prove otherwise.
I did a proof that brought me to a converging limit to ...
2
votes
1answer
39 views
Showing that a sum diverges
Suppose that $a_{j} \geq 0$ and that $\sum a_{j}$ diverges. Prove that $\sum\frac{a_{j}}{1+a{j}}$ diverges.
The hint that is given is show that it if it converges $a_{j} \rightarrow 0$. I don't ...
1
vote
2answers
48 views
Prove or disprove a result for a double sequence.
Suppose that a double sequence $\{a_{n,k}\}=\left\{\frac{1}{n^{\frac{k-1}{k}}}\right\}$. Prove or disprove $\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{k}a_{n,k}=0$.
0
votes
1answer
53 views
Vector space of convergent sequences, prove it's complete
In the space of convergent sequences, such that for a convergent sequence $(a_n)$ we have $\sum _{n=1} ^{+ \infty} a_n ^2 < \infty$, we define a norm $(a_n) \rightarrow \sqrt{\sum _{n=1} ^{+ ...
2
votes
2answers
69 views
Real analysis - converging sequence [duplicate]
My answer
Solution
1).
$Let\; \epsilon = L/2 > 0 \mbox{thus by definition of}\; x_m→L, \mbox{there exists}\; a \;n_o∈ N \;\mbox{such that }∀m>n_o\; \\
|x_m - L|< ε\\
-ε <|x_m - L| < ...
1
vote
1answer
46 views
If $f_n$ converges uniformly to $f$ on a measure space, show integral of $f_n$ converges to integral of $f$.
Please help me with this problem!
Let $(\Omega,\cal F, \mu)$ be a measure space on which $(f_n)$ is a sequence of bounded, measurable, real-valued functions converging uniformly to $f$.
If ...
1
vote
1answer
52 views
bounds for derivatives pass to the limit?
Let $(f_n)_{n\in\mathbb{N}}$ a sequence of $C^1$ functions on $\mathbb{R}$ pointwise convergent to the $C^1$ function $f$.
I know that $f_n'(x)\geq a_n\cdot c(x)$ for all ...
5
votes
2answers
65 views
Which is the function that this sequence of functions converges [duplicate]
Prove that $$ \left(\sqrt x, \sqrt{x + \sqrt x}, \sqrt{x + \sqrt {x + \sqrt x}}, \ldots\right)$$ in $[0,\infty)$ is convergent and I should find the limit function as well.
For give a idea, I was ...
2
votes
2answers
63 views
Show uniform convergence of bounded functions implies uniform boundness.
Suppose that $f_{n} : E\rightarrow \mathbb{R}$ is a sequence of bounded functions that converge uniformly. Prove that there exists $M > 0$ such that for all n in $\mathbb{N}$ and $x$ in $E$,
...
5
votes
2answers
105 views
Need to prove $f$ continuous at $x_0$ iff for every monotonic sequence $(x_n)$ converging to $x_0$ we have $\lim f(x_n)=f(x_0)$
This was a problem that the Professor went over in class, but I am having trouble understanding and finishing the proof. The full question is:
$f:I \rightarrow \mathbb R$ is continuous at $x_0 \in I$ ...
1
vote
2answers
46 views
Convergent sequence question…
I have a homework question that I am not sure how to begin. We are asked to suppose $\{a_n\}$ and $\{b_n\}$ are sequences such that $\{a_n^2 + b_n^2\} \rightarrow 0$.
We have to prove $\{a_n\} ...
0
votes
1answer
22 views
Understanding this theorem about continuity at $c$ and a sequence converging to $c$
I want someone to explain to me just this part:
Let $f:D\rightarrow \mathbb{R}$ and let $c\in D$. Then $f$ is continuous at $c$ if and only if, whenever $X_n$ is a sequence in $D$ that converges ...
0
votes
0answers
44 views
Math Analysis - Problem with finding domain of series of functions
a. Find the domain of the definition $D \subset \mathbb R$ for the function
$$f(x)=\sum_{n=1}^\infty (-1)^n{x \over n+x}$$
b. For what values of $x\in D$ the function $f$ is differentiable?
Well I ...
