Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n x^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex (or more generally an element of a Banach algebra).

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2answers
46 views

Calculate the radius of convergence [closed]

Being $\sum _{n=0}^{\infty \:}a_n\cdot x^n$, $a_1=a_0=1$ $a_{n+1}=a_n + a_{n-1}$ show that the radius of convergence is $\dfrac{-1+\sqrt{5}}{2}$ Thanks!
3
votes
2answers
125 views

Calculate the value of $\sum\limits _{n=1}^{\infty }\:\dfrac{n}{2^n}$ [closed]

In a previous question it is asked to represent $f(x)=\dfrac{x}{1-x^2}$ as a power series. It gave me $\displaystyle\sum _{n=1}^{\infty \:}x\left(2x^2-x^4\right)^{n-1}$. Then they ask to use the last ...
1
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0answers
28 views

Rational approximation or series expansion of $K_0$ and $K_1$ for small z

I'm looking for a series expansion of the modified Bessel functions of second kind $K_0(z)$ and $K_1(z)$ for $$|z|<5, ~~|phase(z)| < \pi$$ My $z$ can be described as $z = a\cdot \sqrt{ix}$, ...
3
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3answers
65 views

how to represent $\int \frac{\arctan \left(x\right)}{x}dx$ as a power series?

I have no idea. I don t even no how to calculate the primitive can you help me?
3
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2answers
44 views

How to represent $\ln(5-x)$ as a power series?

I know that $$ \ln(1+x)=\sum _{n=1}^{\infty }\:\left(-1\right)^{n-1}\frac{x^{n}}{n} $$
0
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2answers
27 views

finding the residue of the following

I can find $$Res_{z=0} \frac{\sin z}{z^4} $$ but stuck with finding $$Res_{z=0} \frac{\cot z}{z^4} $$ so please help me
0
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2answers
42 views

By completing the square, show that $\int_{0}^{\frac{1}{2}}\frac{dx}{x^2-x+1}=\frac{\pi }{3\sqrt{3}}$

By completing the square, show that $\int_{0}^{\frac{1}{2}}\frac{dx}{x^2-x+1}=\frac{\pi }{3\sqrt{3}}$. I found that $\int_{0}^{\frac{1}{2}}\frac{dx}{x^2-x+1}$ equals to ...
0
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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 ...
3
votes
2answers
36 views

Help with understanding a proof about differentiating a real power series

I'm stuck trying to understand a proof of the following theorem: Let $ \sum a_nx^n$ be a power series with radius of convergence $ R $. Then $ \sum na_nx^{n-1}$ also has radius of convergence $ R $. ...
0
votes
3answers
77 views

If $f(x) = e^{x^{2}}$, show that $f^{(2n)}(0)=(2n)!/n!$ [closed]

If $f(x) = e^{x^{2}}$, show that $f^{(2n)}(0)=(2n)!/n!$
4
votes
2answers
45 views

Suppose that $f(x)=\sum_{n=0}^{\infty }c_n{x^{n}}$ for all x

Suppose that $f(x)=\sum_{n=0}^{\infty }c_n{x^{n}}$ for all x Show that if $f$ is an odd function, $c_{0}=c_{2}=...=0$ and if $f$ is an even function, $c_{1}=c_{3}=...=0$
3
votes
4answers
96 views

Sum the following $\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)} $

Evaluate: $$\sum_{n=0}^{\infty} \frac {(-1)^n}{4^{4n+1}(4n+1)} $$ I rewrote the sum as $$\sum_{n=0}^{\infty} \frac {1}{4^{8n-7}(8n-7)} - \sum_{n=0}^{\infty} \frac {1}{4^{8n-3}(8n-3)}$$ Now, I ...
2
votes
0answers
78 views

Sum of $k$th power of first $n$ natural number (power sum)

I was working on a problem which involves computation power sum (summation of $k^{th}$ power of first natural number), can someone help me how to simplify the below equation. I can compute power sum ...
0
votes
2answers
68 views

Prove $f(x)=\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k $ defines a continuous function on $\mathbb{R}$.

Prove $$f(x)=\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k$$ defines a continuous function on $\mathbb{R}$. I think we can show that if $\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k $ is uniformly ...
1
vote
2answers
81 views

radius of convergence $\displaystyle \sum_{n=1}^{\infty}n! x^{n!}$

I just wondering radius of convergence following series $$ \sum_{n=1}^{\infty}n! x^{n!} \\ $$ My 1st attempt is 'root test' $$ \sqrt[n!]{|a_{n!} |} =\sqrt[n!]{|n! |} =\sqrt[t]{t} \rightarrow 1 $$ So, ...
-1
votes
2answers
34 views

Find the sequence $\{c_n\}$ for $c_n = \alpha \cdot c_{n-1} + {\alpha}^{\beta-n}$

Let $\alpha$ and $\beta$ be two given constants, how to find the sequence $\{c_n\}$ for $c_n = \alpha \cdot c_{n-1} + {\alpha}^{\beta-n}$, where $c_0 = {\alpha}^{\beta}$.
0
votes
3answers
30 views

Calculate the area of region between curve f and x axis using series

Consider $f(x)=\sqrt{1+x^4}$ I need to approximately calculate the area of a region between a curve $f$ and the x-axis on [0,1]. However, I need to do this using the five first term non-null of the ...
3
votes
0answers
70 views

Series that diverges at infinitely many points on the unit circle

Initial problem Given $A=\{\alpha_1,\dots,\alpha_k\}$ with $|\alpha_i|=1$, does there exist a power series $\sum a_nz^n$ that converges everywhere on the unit circle except when $z\in A?$ ...
0
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4answers
67 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.
2
votes
1answer
39 views

Find the radius of convergence of the power series $J_{o}(x)= \sum_{n=0}^{\infty}\frac{(-1)^nx^{2n}}{2^{2n}(n!)^2}$

Find the radius of convergence of the power series $$J_{o}(x)= \sum_{n=0}^{\infty}\frac{(-1)^nx^{2n}}{2^{2n}(n!)^2}$$ Should I separate this into the product of two limits, namely ...
4
votes
1answer
78 views

Is there a generalization of the fundamental theorem of algebra for power series?

Given the similarity between polynomials and power series, I was wondering if there is any generalization of the fundamental theorem of algebra for power series. I understand that it doesn't make much ...
4
votes
1answer
46 views

Limit of ratio between a power series and a “subset” of the power series

$B$ is an infinite power series that converges everywhere, and $A$ is an infinite power series that converges everywhere which is composed only of terms found in $B$ - both have nonnegative real ...
12
votes
1answer
243 views

Prove that $\exp(\log(\frac{1}{1-x})) = \frac{1}{1-x}$

I am trying to prove this directly by comparing the coefficients in the two series rather than using formal calculus. Here is what I have so far, but I think I made a mistake: \begin{align*} ...
2
votes
0answers
103 views

Prove that periodic analytic function can be written as $\sum_{-\infty}^{\infty} c_n e^{2\pi inz}$

This question involves the following homework problem: PROBLEM Suppose $f$ is analytic in the upper half plane and periodic of period 1. Show that $f$ has an extension of the form ...
0
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1answer
33 views

Convergence of a power series at points where ratio test is inconclusive

I need to find the interval of convergence of the following power series using either the ratio test , integral test or comparison test. Using the ratio test I found that it will converge for $ -4 ...
3
votes
1answer
27 views

Radius of Convergence for a Complex Function

I'm really rusty on my series convergence, and I guess I was more asking for a bit of clarification on a question I'm working on. I've been asked to find the power series expansion for $\frac{1}{3 - ...
0
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0answers
29 views

Radius of Convergence and the Frobenius Method

Consider the equation $$4xy'' + 2y'+ y = 0$$ I know that $x=4$ is a regular singular point, and in the notation that my uni uses, we say that: $$(x-x_0)^2 y'' + (x-x_0)p(x)y' + q(x)y = 0$$ where ...
0
votes
2answers
56 views

Sum of potencies with higher potency as clue

I am supposed to calculate the following as simple as possible. Calcute: $$1 + 101 + 101^2 + 101^3 + 101^4 + 101^5 + 101^6 + 101^7$$ Tip: $$ 101^8 = 10828567056280801$$ I have absolutely no idea how ...
1
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2answers
32 views

a series derived from a holomorphic function converges implies that the coefficients converge to $0$

Let $D=\{z\in\mathbb{C}\mid |z|<2\}$. Let $f:D\setminus\{\frac{i}{2}\}\longrightarrow \mathbb{C}$ be holomorphic with $f(z)=\sum_{n=0}^\infty a_nz^n$ for any $|z|<\frac{1}{2}$. Suppose $a_n\neq ...
1
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0answers
43 views

Integration vs Summation

I am interested in how one might generally evaluate, or estimate $$G(x)=\sum_{n=1}^{\infty}f(n)x^n-\int_{0}^{\infty}f(t)x^tdt$$ as $x\to1^-$, and for a continuous $f$, and such that the integral ...
2
votes
1answer
48 views

Radius of convergence

(a) Determine the radius of convergence to the power series $f(x)= \displaystyle\sum\limits_{n=0}^\infty \frac{(2n)!}{(n!)^2}x^n$. Should I use the ratio test? (b) Assume the validity of the ...
4
votes
1answer
52 views

Some computation with a power series

This is an old qualifying exam problem that has me stumped. Suppose $$F(x):=\sum_{n=0}^{\infty}a_n x^n$$ converges in some neighborhood of the origin. I want to compute $$\sup\left\{\delta>0 : ...
0
votes
1answer
64 views

Question about infinitely many times differentiable function.

Could you please give me some hint how to solve this problem: Suppose $f(x)$ is infinitely many times differentiable function on R, $f(0)=f'(0)=f''(0)=0$. Prove : for all $A>0$ exists some ...
2
votes
2answers
78 views

Sum of products of binomial coefficient $-1/2 \choose x$

I am having trouble with showing that $$\sum_{m=0}^n (-1)^n {-1/2 \choose m} {-1/2 \choose n-m}=1$$ I know that this relation can be shown by comparing the coefficients of $x^2$ in the power series ...
0
votes
2answers
70 views

Radius of Convergence…very tricky question using very little information

Even one of the maths gurus at my university struggled to get a proof out for this... so I'm almost completely lost! This is the question: Let $a_{n}$ be a sequence of positive real numbers for which ...
0
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1answer
33 views

Do I have mistakes in my calculations (power series, convergence)?

I'm not sure I got all of these problems right. I'd really appreciate any sort of feedback. For which $x \in \mathbb{R}$ do the following series converge? Problem 1 For ...
0
votes
0answers
66 views

Taylor theorem remainder term

I'm having trouble applying the formula for the remainder in the Taylor's theorem. From Wikipedia we know that for $f(x)=f(a)+f'(a)(x-a)+…\frac{f^{(n)}(a)}{n!}(x-a)^{n}+R$ the remainder $R$ in the ...
6
votes
2answers
164 views

Sum of sum of $k$th power of first $n$ natural numbers.

I was working on a problem which involves computation of $k$-th power of first $n$ natural numbers. Say $f(n) = 1^k+2^k+3^k+\cdots+n^k$ we can compute $f(n)$ by using Faulhaber's Triangle also by ...
0
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1answer
14 views

Some questions about series and generalized integrals

Do someone of you know any good site about generalized integrals and series (convergence and divergence) with example exercises with detailed solutions ? I would be happy if you did know any. My ...
4
votes
1answer
49 views

Complex power series divergent and convergent on dense subsets of the circumference of convergence?

Is it possible to have a complex power series $ \sum a_nz^n $ with radius of convergence R such that the series diverges on a dense subset of the circumference of convergence and converges on another ...
2
votes
1answer
42 views

Two questions about this solution /proof

Consider the following theorem: If $\sum_{n=0}^\infty a_n x^n $ converges for all $x \in (-R,R)$ then the differentiated series $\sum_{n=0}^\infty n a_n x^{n-1}$ converges for all $x \in (-R,R)$. ...
4
votes
1answer
59 views

When is a Fourier series analytic?

By Fourier theory, every continuously differentiable function $f : S^1 \to \mathbf C$ admits a unique, uniformly convergent Fourier expansion $$f(\theta) = \sum_{n\in \mathbf Z} a_n e^{in\theta}.$$ ...
4
votes
0answers
58 views

About $\prod{\left(1-q^n\right)^{5}}$

Is there a result about the non-vanishing of coefficients of $$\prod_{n=1}^{+\infty}{\left(1-q^n\right)^{5}}=1-5q+5q^2+10q^3-15q^4-6q^5-5q^6+25q^7+15q^8-20q^9+\cdots \text{ ?}$$ Thanks !
1
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0answers
37 views

Proof of pointwise convergence of derivative of power series

I proved: If $\sum_{n=0}^\infty a_n x^n$ converges for all $x \in (-R,R)$ then the differentiated series $\sum_{n=1}^\infty na_n x^{n-1}$ converges for all $x \in (-R,R)$. Please could somebody tell ...
0
votes
1answer
35 views

Given a power series

Let c be a fixed number and consider the power series $\displaystyle\sum_{n=1}^ \infty \frac{c^{n-1}}{n} x^{n}$. a) Determine the convergence radius r for every value of $c \in \mathbb{C}$. In this ...
3
votes
0answers
63 views

Proof of Abel's theorem

I tried to prove: If $g(x) = \sum_{n=0}^\infty a_n x^n$ is a power series that converges at $x= R > 0$ then it converges uniformly on $[0,R]$. Please can you check my proof? Let $\varepsilon ...
3
votes
0answers
47 views

Is there an algebraic description of the ring of analytic functions on the real projective line?

Apologies for the long question. Let $X=\mathbf P^1(\mathbf R) \subseteq \mathbf P^1(\mathbf C)$ be the real projective line. Let $\mathcal O_X$ be the sheaf of real-analytic complex-valued functions ...
2
votes
3answers
20 views

Absolute sequence unbounded within radius of convergence

Let $R$ be the radius of convergence of the complex power series $a_nz^n$ with $0<R<\infty$. Show that if $|z|>R$, then the sequence $|a_nz^n|$ is unbounded. Trying by contradiction: So ...
1
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0answers
25 views

How do we show $\ln z=\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}(z-1)^n$ for all $z\in\mathbb{C}$ with $|z-1|<1$?

Let $$g:B_1(1):=\left\{z\in\mathbb{C} :|z-1|<1\right\}\to\mathbb{C}\;,\;\;\;z\mapsto\ln z-\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}(z-1)^n$$ (1) In a first step, I'm asked to show, that $g$ is ...
0
votes
1answer
15 views

Limit when $y>>a$ of a derived solution

I am able to do part d), however I am very stuck on part e). If $y >> a$ then surely we get $\phi(x,y)$ $= \frac{1}{\pi} \Big[ tan^{-1} \Big( \frac{x+a}{y}\Big)-tan^{-1} ...