Tagged Questions

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
21 views

Which solution is the right one??

If we want to solve the equation $sec^2(x)$ for finding the all roots(real and complex), we have two ways: 1-Direct solving for $sec^2(x)=0$ 2-Or by convert the above equation to polynomial series as ...
1
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2answers
28 views

Is $-\log (1-z) = \sum_{n=1}^{\infty}\frac{z^n}{n}$ for $z \in \mathbb{C}, \|z\|=1, z \neq 1$?

Is $-\log (1-z) = \sum_{n=1}^{\infty}\frac{z^n}{n}$ for $z \in \mathbb{C}, \|z\|=1, z \neq 1$ ? In any case, why?
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0answers
30 views

Show that $\sum_{n=1}^{\infty} \frac{z^n}{n}$ converges for $z \in \mathbb{C}$ such that $\|z\|=1$ but $z \neq 1$

I know I could use Dirichlet's test, but I am wondering if the Taylor series of $- \ln (1-z)$ can be used in some way to prove it for $\|z\|=1$, $z \neq 1$. I know the convergence radius is 1 so it is ...
1
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4answers
36 views

Convergent complex series

Is $$\sum\limits_{n=1}^\infty \frac{i^n}{n} $$ convergent? Im confused as to how to solve this question, I've been trying to use ratio test but that doesn't seem to be helping.
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2answers
26 views

Integrated series identity with Legendre Polynomials

The Legendre Polynomials can be defined in many different ways and have several properties. Many of these can be found in books or in the net, but I couldn't find this one anywhere: Prove that: ...
0
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1answer
37 views

Difference between power series method and Frobenius method

There is the power series method for solving ordinary differential equations: one looks for solutions of the form $\sum c_n x^n$, and derives algebraic relations between coefficients $c_n$. Then ...
0
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2answers
40 views

Calculate the power series

I want to find the power series of $\frac{1}{3!}$ in the field $\mathbb{Q}_3$. In order to do this, do I have to solve the congruence $3!x \equiv 1 \pmod{3^n} \Rightarrow 6x \equiv 1 \pmod 3$? If ...
0
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1answer
20 views

Is this method of finding range of x for which given series is convergent, wrong?

There was this question in our midsem question paper: We had to find out the range of values of x (x is positive) for which the given series is convergent, Given series was $\sum_{n = 1}^{\infty} (a ...
5
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1answer
44 views

Can a power series converge uniformly on $(-1,1)$ but not on $[-1,1]?$

I am taking a course in analysis, and I am wondering whether it possible for a power series with radius of convergence $1$ to converge uniformly on $(-1,1)$ but not on $[-1,1]?$ I don't think this ...
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2answers
37 views

Prove that the limit of a series, containing 1/{powers-of-2}, is not rational

I have a series, $$x_n = \sum_{k=0}^n2^{-k^2-k}, \forall n \in N$$ I have to find it's limit and prove it is not in Q(it is not rational). I tried to write it $x_n=1+\frac{1}{2^1*2^1}+\frac{1}{2^4* ...
2
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0answers
27 views

How to compute the following sum?

How to compute the following sum? $$\sum_{k=1}^{\infty} \frac{k^{k-1} \cdot e^{-k}}{k!}$$ It is likely to be equal $1$ (there is an argumentation that goes back to random graphs). Moreover, i think ...
0
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0answers
16 views

Power series convergent at exactly one point of unit circle [duplicate]

I have to give an example of power series with radius of convergence equal 1 that is convergent at exactly one point of unit circle and divergent at all other points of that circle.
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0answers
34 views

Expanding in powers of $\epsilon$ and big O notation

I do not understand how to approach (D.1) equation Where did that big O notation come from?Is it using taylor series and linear approximation? Thanks in advance
5
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1answer
47 views

Differentiation under the integral sign and counting measure

Consider a power series $f(x)=\sum_{n=1}^\infty a_nx^n$, and assume that $\displaystyle R=\lim_{n\to \infty} \frac{a_n}{a_{n+1}}$ exists. Use differentiation under the integral sign to show that ...
1
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1answer
34 views

notation for first and second derivatives of a power series

I have a power series $$\sum_{k=0}^\infty\frac{c_k}{k!}x^k$$ where $c_k$ is an arbitrary $k$-th term of some sequence. Then ...
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0answers
6 views

Differentiational equation construct power series expansion

I got a question In order to improve the accuracy of your numerical estimate you are to use a power series expansion of y(x)to estimate y(1). (You may find it easier if you multiply both sides of ...
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1answer
27 views

Radius of convergence of $\sum a_n x^{n^2}$

Let $\sum a_n x^n$ be a power series with radius of convergence $R$. What is the radius of convergence of $\sum a_n x^{n^2}$? Can anyone help me here?
2
votes
2answers
25 views

Find the Power Series

How would one write $f(z) = \frac{1}{1-wz}$ as a power series? ( Where $z,w$ are in $C$.) Would it just be $\sum_{n=0}^{\infty} (zw)^n$?
2
votes
0answers
28 views

Taylor polynomial converging pointwise but not uniformly?

Many standard examples of Taylor series $(\exp(x), \sin(x), \cos(x))$ converge uniformly, others don't converge to its original function at all, e.g. $\exp(-x^{-2})$. I couldn't think of any smooth ...
1
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2answers
33 views

If $\sum_{n=0}^\infty c_n x^n$ is convergent for $x=-3$ [closed]

Is the following True or False: If $\sum_{n=0}^\infty c_n x^n$ is convergent for $x=-3 \implies:$ a) $\sum_{n=0}^\infty c_n 2^n$ converges. b) $\sum_{n=0}^\infty c_n 3^n$ converges. EDIT: I found ...
0
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0answers
24 views

Writing solution to an arbitrary ODE with arbitrary initial values as the sum of a power series?

Let $f(t), g(t)$ be polynomials, and let $y$ be a function of $t$. Given the ODE $y'' + f(t) y' + g(t) y = 0$ with initial conditions $y(0) = \alpha$ and $y'(0) = \beta$, write $y$ as the sum of a ...
0
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1answer
21 views

Radius of convergence of $2^n+3^n, n \geq 1$

Find the radius of convergence of the power series where, $a_n= 2^n+3^n, n \geq 1$. The answer is given to be 1. The tests I can use are Cauchy Hadamard Test and Ratio Test. My attempt: Using Ratio ...
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votes
1answer
22 views

Radius of convergence of $1+3x+\frac{3^2x^2}{2!}+\cdots$

The question is to find the radius of convergence of the power series $1+3x+\frac{3^2x^2}{2!}+\frac{3^3x^3}{3!}+\cdots$ The answer is given to be $\frac{1}{3}$ My attempt: $a_n=\frac{3^n}{n!}$ ...
2
votes
0answers
11 views

Derivation of higher order bessel function in terms of lower order functions

I am really stuck trying to prove this.. ((x^-p)Jp(x))’ = -(x^-p)Jp+1(x) ---(1) Can someone please help how to actually prove this step by step, because whichever notes i see, they prove ...
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2answers
15 views

What is $1 + \sum_{k=1}^{\infty} \frac{(it)^k}{k!}a^{2k+1}$?

I want to express $$1 + \sum_{k=1}^{\infty} \frac{(it)^k}{k!}a^{2k+1}$$ in terms of standard functions (exp, cos, sin, etc.), but I just don't see what this function is. Does anybody here have an ...
0
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0answers
66 views

Why is such a the series algebraic but rational?

The coefficients of the series expansion of the algebraic function $A=\frac{1-\sqrt{1-8x^2}}{4x}$ are all intergers: $$A(x)=x+2x^3+8x^5+\cdots$$ But according to Polya's research,if $ F(x)$ is a ...
1
vote
1answer
26 views

Recognising that $\sum_{n=0}^\infty \frac{a^2-b^2(2n+1)^2}{(a^2+b^2(2n+1)^2)^2}=-\frac{\pi^2\mathrm{sech}^2\left(\frac{a\pi}{2b}\right)}{8b^2}$

So I know from Mathematica that: $$\sum_{n=0}^\infty \frac{a^2-b^2(2n+1)^2}{(a^2+b^2(2n+1)^2)^2}=-\frac{\pi^2\mathrm{sech}^2\left(\frac{a\pi}{2b}\right)}{8b^2}$$ I am wondering how someone could ...
0
votes
1answer
27 views

Radius of convergence: $\sum_{k=1}^\infty \frac{x^{2k-1}}{2k-1}$

It is asked to find the radius of convergence of the series $$\sum_{k=1}^\infty \frac{x^{2k-1}}{2k-1}$$ i.e, to find the values of x such that this series converges. Clearly, I could directly apply ...
2
votes
1answer
26 views

Convergence of $\sum(-1)^k\frac{(\ln k)^p}{k^q}$ where $p,q$ in positive $\mathbb{R}$

For any $p, q$ in positive $\mathbb{R}$ $$\sum_{k=2}^{\infty}(-1)^k\frac{(\ln k)^p}{k^q}$$ I want to Use alternative series test for convergence but I'm struggling to verify that $\frac{(\ln ...
0
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0answers
30 views

statements of matrix analysis

Let $y$ be fixed value. Let $A=a(x,y)$ be a matrix and $f_{t}(x)=\frac{\sum_{n=0}^{\infty}{a^{(n)}(x,y)(\frac{1}{t})^n}}{\sum_{n=0}^{\infty}a^{(n)}(y,y)(\frac{1}{t})^n}$ Show that ...
1
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0answers
31 views
+100

Twisted logarithm power series

I recently encountered a power series similar to the one of the $\log(1-x)$ of the form $$ F(x)= \sum_{n=1}^\infty \frac{\psi(n)x^n}{n}, $$ where $\psi$ is some Dirichlet character. Has anyone here ...
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0answers
33 views

Calculating the power series expansion about pi/2 of g(z)=tan[z/2]

Calculating the power series expansion about pi/2 of g(z)=tan[z/2]. Now calculate the expansion about 0. I'm having trouble doing this. I'm not even sure which is the best way to approach it, for ...
5
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0answers
78 views

Can a Power Series tell when to stop?

The naive description of the radius of convergence of a complex power series is as the largest radius so that the ball avoids poles and branch cuts. This makes sense in a world where analytic ...
1
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2answers
24 views

how to find power series in closed form

find a "closed form" of summation n=2 to infinity n(n-1)x^n. I don't have much clue to solve this can anyone please explain how to approach this?
1
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0answers
77 views
+50

What more can i do to this infinite sum?

This question sprung out from another post of mine that was in part by Semiclassical, he Proved the Following: $$ \sum_{n=0}^{\infty} {}_2F_1(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};1)/n! = 2\pi ...
0
votes
1answer
21 views

asymptotic series for “stable distribution”

I'm trying to understand how to get from one equation to another in a certain paper I am studying (DOI:10.1080/00018738100101467, eqs. 4.34 and 4.35). The equations are pretty self contained, so I'm ...
0
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1answer
19 views

Prove the equality with power series

I have to prove for $|x| < 1$ that $$ \ln\frac{2(1-\sqrt{1-x})}{x} = \frac 12 \cdot \frac x2 + \frac 12 \cdot \frac 34 \cdot \frac{x^2}{4} + \frac 12 \cdot \frac 34 \cdot \frac 56 \cdot ...
2
votes
0answers
28 views

Help understand part of the proof. Radius of convergence is $\frac{1}{\limsup |a_n|^{1/n}}$

Can you help me understand the highlighted parts of the proof. Thanks :) Theorem: Let $\sum{a_nz^n}$ be a power series, let r be its radius of convergence. Then $\frac{1}{r} = \limsup |a_n|^{1/n}$. ...
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0answers
16 views

Explain $z(\cos z -1)$ serie expansion

Look at the following expansion, which should be an expansion from for the coefficients $a_0, a_1, a_2, a_3$ $$\begin{align} z(\cos z -1) &= z \left( 1 - z^2/2!+ z^4/4! - z^6/6! + ...
1
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0answers
29 views

Ideal in power series ring

Let $J$ be an ideal in $k[[x_1,...,x_n]]$ such that $(x_{1},...,x_{n})^{2}\subseteq J$, $\{x_{1},...,x_{r}\}\nsubseteq J$ and $\{x_{r+1},...,x_{n}\} \subseteq J$, for some $1\leq r \leq n$. I want to ...
0
votes
2answers
35 views

Cannot expand $\sin(2x^2-4x+3)$ at $x_0 = 1$

Trying to expand $\sin(2x^2 - 4x+3)$ at $x_0 = 1$ to the $O(x-x_0)^n$. After substitution $t = x - 1 $, the problem becames $$\sin(2t^2+1) \text{ at } t_0 = 0$$ While we know that $$\sin(s) = ...
1
vote
2answers
44 views

Estimating the behavior for large $n$

I want to find how these coefficients increase/decrease as $n$ increases: $$ C_n = \frac{1}{n!} \left[(n+\alpha)^{n-\alpha-\frac{1}{2}}\right]$$ with $\alpha=\frac{1}{br-1}$ and $0\leq b,r \leq 1$. ...
1
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0answers
9 views

Simplifing a Cauchy product to find the recurrence relation when solving a differential equation using a power series solution.

I'm having trouble finding the recurrence relation of the following non linear differential equation: $y''(x)+p(x)y'(x)+y^2(x)=0$ with $y(0)=1$ and $y'(0)=0$ where ...
1
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1answer
22 views

Vanishing of Taylor series coefficient

I am solving previous year question paper some competitive exam. Give me some hint to solve the following problem. Let $f$ be an entire function. Suppose for each $a \in \mathbb{R} $ there exists at ...
1
vote
2answers
24 views

Importance of the first term in a Taylor series

Suppose you have a function $f(x)$ whose Taylor series can be represented as the power series $$a_0 + a_1x^2+a_2x^4+...$$ If you are told that for $x\in\mathbb{R}_+$, $$a_0 + a_1x^2 + a_2x^4 + ...
1
vote
3answers
39 views

Compute the following sum for any x?

Compute the following sum for any x? $\sum_{n=0}^\infty {(x-1)^n\over (n+2)!}$ I am having trouble to compute that sum. It looks like geometric series but I don't know where to start. Can everyone ...
1
vote
1answer
24 views

Finding a Taylor Series representation of $f(x)=\ln(\frac{1+2x}{1-2x})$ centered at $0$.

I'm trying to find a Taylor Series representation of $f(x)=\ln(\frac{1+2x}{1-2x})$ centered at $0$. So I am using the Maclaurin Series representation of $f(x)=\ln(1+x)$ which is ...
2
votes
3answers
123 views

Estimating the series: $\sum_{k=0}^{\infty} \frac{k^a b^k}{k!}$

Any idea on how to estimate the following series: $$\sum_{k=0}^{\infty} \frac{k^a b^k}{k!}$$ where $a$ and $b$ are constant values. Greatly appreciate any respond.
0
votes
0answers
21 views

Expression for a series of squared sines

Does anyone know if there is a single expression for $$-\frac{1}{2}\sum_{j=1}^{\infty}\sin^2\left(\frac{2\pi x}{3^j}\right)$$ or at least a nicely-expressed upper bound? I've already computed that ...
1
vote
4answers
106 views

Power series in $\mathbb{Q}_5$

Could you help me to find the first five positions of the power series in $\mathbb{Q}_5$ of $\frac{1}{2}$? How can I do this? Is there a general formula?