Tagged Questions
0
votes
1answer
33 views
power series quotient of polynomial functions
I have given $g(x)=\sum_{k=1}^\infty k^2x^k$. Why can you now write $g:(-1,1)\rightarrow\mathbb R$ as a quotient of two polynomial functions?
I just know the radius of convergence is ...
2
votes
1answer
38 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
33 views
Points around which one expands and the radiuses of convergence
I'm trying to make sense of the following passage:
Let $f(x)=\frac{1}{x+1}$ and $R_0$ the radius of convergence of the Taylor series of $f$ around $x_0=0$, analogously: $R_1$ — around ...
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
$$
...
0
votes
2answers
49 views
Ratio test and the radius of convergence
Let
$$
\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|
$$
exists (the limit exists and is a real number), it is the ...
3
votes
2answers
73 views
May I use the triangle inequality for infinite series?
I have to prove the following statement:
Let $\lim_{n\to \infty}r_n=0$. Show that $\forall\varepsilon>0 \ \ \ \exists \, n_0 \in \mathbb N \ \ \ \forall x \in(-1,1):$
$$\left\lvert ...
0
votes
0answers
72 views
Is there a power series that converges to the function $f(x)= \lvert x\rvert$ for all $x$?
Is there a power series that converges to the function $f(x)= \lvert x\rvert$ for all $x$?
I am pretty lost on how to even start this.
2
votes
2answers
46 views
Analytic functions of a real variable which do not extend to an open complex neighborhood
Do such functions exist? If not, is it appropriate to think of real analytic functions as "slices" of holomorphic functions?
2
votes
1answer
72 views
Show that some $C^\infty$ real function is analytic
Consider the function $f:(a,b) \rightarrow \Bbb R $, which is $C^\infty$ class. It is required to show that if the function $f$ satisfies: $\forall_{n\in\Bbb N_+} \forall_{x\in(a,b)} f^{(n)}(x) \ge 0 ...
2
votes
2answers
112 views
$\sum (-1)^n/n$ fails the p-series test, but passes the alternating series test?
P-Series Reference
Alternating Series Test Reference
$$ \sum_{i=0}^\infty \frac{(-1)^n}{n} $$
This alternating series fails the p-series test because the exponent of n = 1.
Yet it seems to pass ...
0
votes
3answers
42 views
Power Series Proof w/ Binomial Coef.
Prove that, for any positive integer k,
$\sum_{n=0}^\infty {{n+k \choose k}z^n}$ = $1/(1-z)^{k+1}, |z| < 1$
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,
$$R=\lim_{n\to\infty} \frac{[1\cdot 3 \cdots ...
1
vote
2answers
35 views
expressing $\int tan^{-1}tdt$ as a power series
I have a question which asks me to state $\int_0^x\tan^{-1}(t)dt$ as a power series in $x$ and then use that result to show that $\frac{\pi}{4}-\log\sqrt{2}= 1-1/2-1/3+1/4+1/5--++\cdots$
For the ...
0
votes
2answers
41 views
Power series coeffieients
Determine the coefficients of the power series that defines a function with the following
properties: $f''(z) = −f (z), f (0) = 1, f'(0) = 0.$
1
vote
1answer
51 views
Taylor expansion with integral?
I have looked at a version of a Taylor expansion that has an integral- for the first time. Is this the same as the usual version of a Taylor expansion without integrals? Also, do the $\alpha's$ have ...
7
votes
1answer
108 views
Does a convergent power series on a closed disk always converge uniformly?
If I have a power series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i} \in\mathbb{C}[[z]]$ with radius of convergence $r>0$ and I know that the series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i}$ ...
1
vote
1answer
40 views
Power series and power series expansion
I am looking for help with a problem. Here is the question I am working on:
Consider the power series
$$\sum_{n=1}^{\infty}(-1)^n \frac{n+1}{n}x^{n}$$
(a) Determine the radius of ...
1
vote
1answer
73 views
Where are power series uniformly continuous?
As far as I know, $f(x)=\sum\limits_{n=0}^\infty a_n(x-x_0)^n$ is continuous on the whole convergence interval $K:=\{x\in\mathbb R:|x-x_0|<r\}$. Is there anything we could say about uniform ...
1
vote
1answer
64 views
Showing that if $|a_n|<|b_n|$ and $\sum b_k x^k$ converges, then $\sum a_k x^k$ converges
Hypothesis: $|a_{n}|<|b_{n}|$ for all natural $n$, and $\displaystyle\sum_{k=0}^{\infty} b_{k}x^k$ converges on $(-R, R)$. Prove that $\displaystyle\sum_{k=0}^{\infty} a_{k} x^k$ converges on ...
3
votes
2answers
83 views
Power series, derivatives, integrals, and different intervals of convergence
I am working on the question below. It involves finding three different power series that meet certain conditions.
(a) Find a power series $\sum_{n=0}^{\infty} a_nx^{n}$ that has a different ...
2
votes
5answers
77 views
Power Series and Radius of Convergence Question
I've got a start on the question I've written below. I'm hoping for some help to finish it off.
Suppose that the power series $\sum_{n=0}^{\infty}c_n x^n$ has a radius of convergence $R \in (0, ...
0
votes
1answer
52 views
Can a power series always be integrated term-by-term inside the circle of convergence of its sum function?
Is it true that a power series can always be integrated term-by-term inside (i.e. in the interior of) the circle of convergence of its sum/limit function?
My complex analysis textbook merely states ...
2
votes
2answers
42 views
Partial Fractions - combinatorics - trouble
Having a very hard time with this question:
Q: Use partial fractions to find the power series expansion of $$\frac{1+5x}{1-2x-3x^2}$$
2
votes
1answer
31 views
Specific question about the consequence of composing power series
Please bear with my possible abuse of notation/terminology. Consider the power-series composition f(g(x)). If g's range lies within f's interval of convergence, and if series g has a constant term 0, ...
0
votes
1answer
83 views
Using the general binomial theorem to find a series-like expression for $\sqrt 2$
How do I use the general binomial theorem (i.e. the series expansion of ${(1+x)^\alpha}$ for $ |x|<1$) to show the following? $$\sqrt 2=1+\frac 1{2^2}+\frac{1\cdot3}{2!\cdot{2^4}} ...
9
votes
3answers
155 views
Can the the radius of convergence increase due to composition of two power series?
When composing power series, is the radius of convergence the minimum of that of the individual series, or is it like for multiplication and addition of power series where the resultant radius of ...
3
votes
4answers
193 views
Show that $\displaystyle\lim_{n\to\infty} nx^n=0$ for $x\in[0,1)$.
Show that $\displaystyle\lim_{n\to\infty} nx^n=0$ for $x\in[0,1)$.
3
votes
2answers
152 views
Proof of the “Radius of Convergence Theorem”
I can't figure out how it is valid to invoke the Absolute Convergence Theorem, whose hypothesis is "Let the power series have radius of convergence R", to establish case c of the Radius of Convergence ...
9
votes
1answer
97 views
Is this (classical?) exercice missing a hypothesis?
A friend just told me about an exercice he was given quite a few years ago, but he wasn't sure wether he remembered all the hypothesis correctly. Does anybody recognize this?
Let $f$ be a smooth ...
1
vote
0answers
50 views
inequality for series
Let $j \in Z_+$. Set
$$
a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!}
$$
and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$.
Let $X(i)=|a^{(2i)}_j|j!$. Verify that $X(i)\leq X(1)$ for ...
1
vote
1answer
82 views
What is the radius of convergence?
$\displaystyle\sum_{n=1}^{\infty}x^{2n-1}/a_{n}$
What is the radius of convergence?
Ps: I found that $\limsup|1/a_{n}|^{1/n}=1/6$
But I am confused because of $x^{2n-1}$
What is the radius of ...
1
vote
1answer
18 views
What values does the function $Z(y)$ have at various interval?
When $y\leq0$; $H(y)=0$
When $y>0$; $H(y)=e^{-\dfrac1y}$
What values does the function $Z(y)$ have at various interval? Where $Z(y)=H(1-y)(1+y)$
Please show this!
3
votes
2answers
70 views
Find the radius of the series
$$\sum_{n=1}^{\infty}\frac{x^{n}}{n^{2}{(5+\cos(n\pi/3))^{n}}}$$
What is the radius of the convergence of the series?
Please show clearly and help me how to solve this. Thank you!
I know the ...
1
vote
2answers
91 views
Radius of convergence is 1
Let's assume that $\displaystyle\sum_{n=0}^{\infty}a_{n}$ is convergent conditionally. Then prove that the radius of convergence of $\displaystyle\sum_{n=0}^{\infty}a_{n}x^{n}$ is equal to $1$
Please ...
1
vote
2answers
113 views
Show that the radius of convergence of the power series is at least 1
If the coefficients ${a_i}$ of a power series $\displaystyle\sum_{i=0}^{\infty}a_{i}x^{i}$ form a bounded sequence show that the radius of convergence of the power series is at least $1$
How to solve ...
2
votes
1answer
65 views
Let $a_n=\frac{1}{n^{\sqrt n}}$. Determine the exact interval of convergence of the power series $\sum a_nx^n$.
Let $a_n=\frac{1}{n^{\sqrt n}}$. Determine the exact interval of convergence of the power series $\sum a_nx^n$.
This is what we thought:
$\limsup|a_n|^{\frac1n}=1$. Therefore $R=1$, so we only have ...
1
vote
3answers
76 views
How to show that for $|x|<1$, $1+2x+3x^2+\cdots=\frac1{(1-x)^2}$? [duplicate]
I saw this formula and I don't know if it has a certain name or how can it be proven
If $|x|<1$ then $$1+2x+3x^2+4x^3+\cdots= \frac{1}{(1-x)^2}$$
Can someone tell me anything about it? How ...
10
votes
1answer
165 views
Abel limit theorem
I would like to know if the Abel limit theorem works if the limit is infinite.
Let the series $\sum_{k=0}^\infty a_k x^k$ have radius of convergence 1. Assume further that $\sum_{k=0}^\infty a_k = ...
1
vote
2answers
111 views
$f(x) := \exp(- \frac 1 {x^2})$ analytic around $0$ ?
Define $f(x) := \exp(- \frac 1 {x^2})$ and $f(0) := 0$. Is there a power series $\sum c_n (x-0)^n$ which converges to $f$ on some set $(-R,R)$,$R > 0$ around $0$ ?
0
votes
1answer
44 views
Powerseries of $\frac 1 {1-x}$ around $a \in \mathbb R \backslash \{0\}$.
Can someone help me finding a power series around $a \in \mathbb R \backslash \{0\}$ which converges to $\frac 1 {1-x}$ for $|x-a| < R$ and some $R > 0$
?
2
votes
2answers
51 views
radius of convergent of the power series $1$ or convergent for all $x$
$a_n$ be a sequence of integers such that such that infinitely many terms are non zero, we need to show that either the power series $\sum a_n x^n$ converges for all $x$ or Radius of convergence is ...
0
votes
1answer
30 views
Comparison of terms of power series and the functions they represent
Hopefully someone better versed in real analysis than I can help with the following:
If $f(x)$ and $g(x)$ are functions on the real line, and $f_n$ and $g_n$ are the coefficients of their series ...
3
votes
1answer
114 views
The radius of convergence of the power series $\sum_{0}^{\infty}P(n)x^n$
I came across the following problem:
Let $P(x)$ be a non-zero polynomial of degree $N.$ The radius of convergence of the power series $\sum_{0}^{\infty}P(n)x^n$
(a)depends on $N,$
(b)is $1$ ...
3
votes
2answers
64 views
Power series related problem
I came across a problem that says:
It is given that $\sum_{n=0}^{\infty}a_{n}z^{n}$ converges at $z=3+4i.$
Then the radius of convergence of the power series $\sum_{n=0}^{\infty}a_{n}z^{n}$ is
...
2
votes
1answer
86 views
Analytical function taking rationals to rationals.
Suppose $f:I \rightarrow \Bbb R$ is an analytic function defined on the interval $I\subset \Bbb R$ with the property that for every $q \in \Bbb Q:f(q)\in \Bbb Q$. Does this already imply that $f\in ...
1
vote
1answer
111 views
Cauchy product and the exponential function
Simplify the following series using the Cauchy product
$$\sum\limits_{k=1}^\infty\frac{1}{k!}\cdot\sum\limits_{j=1}^\infty\frac{1}{j!}$$
...
12
votes
2answers
354 views
$\lim\limits_{x\to\infty}f(x)^{1/x}$ where $f(x)=\sum\limits_{k=0}^{\infty}\cfrac{x^{a_k}}{a_k!}$.
Does the following limit exist? What is the value of it if it exists?
$$\lim\limits_{x\to\infty}f(x)^{1/x}$$
where $f(x)=\sum\limits_{k=0}^{\infty}\cfrac{x^{a_k}}{a_k!}$ and $\{a_k\}\subset\mathbb{N}$ ...
1
vote
0answers
34 views
How to show that $f(x,y)$ are real analytical
Given a real function $f(x,y)$ on $R^2$, we know that (from wiki) it is called real analytic if it is locally given by a convergent power series.
My question is that whether there has some principle ...
3
votes
0answers
149 views
domain of convergence of a multivariable taylor series
consider the rational function :
$$f(x,z)=\frac{z}{x^{z}-1}$$
$x\in \mathbb{R}^{+}/[0,1]\;\;$, $z\in \mathbb{C}\;\;$ .We wish to find an expansion in z that is valid for all x and z. a Bernoulli-type ...
0
votes
2answers
136 views
Radius of Convergence of “Shifted” Power Series
Suppose that $\sum_0^\infty a_nz^n$ has radius of convergence $1$ and suppose that $|z_0|=r<R$
Let $g(z)=\sum_0^\infty a_n (z-z_0)^n$
Prove that $g(z)$ has radius of convergence at least $R-r$.
I ...
