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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1answer
75 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 ...
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1answer
87 views

Demonstrating the coefficients of the power series of $\frac{1}{1-z-z^2}$ satisfies a recurrence relation.

I have the power series $$\frac{1}{1-z-z^2} = \sum_{n=0}^{\infty} c_nz^n$$ and I'd like to show that the coefficients of this power series satisfy $c_n=c_{n-1}+c_{n-2}$. I thought the most obvious way ...
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1answer
294 views

Is there a real power series with radius of convergence 1 that converges uniformly on (−1,1)?

Is there a real power series with radius of convergence $1$ that converges uniformly on $(−1,1)$? I am guessing the answer is yes, if we can construct a function with power series such that it ...
1
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1answer
80 views

What is the bound of coefficients of the series $e^t$?

I'd like to get a bound of the coefficients of the series: $$e^t = 1 + \frac{t^1}{1!} + \frac{t^2}{2!} + \dots + \frac{t^n}{n!} + \dots$$ In other words, after $n$ terms we get a term ...
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5answers
1k views

Use Cauchy product to find a power series represenitation of $1 \over {(1-x)^3}$

Use Cauchy product to find a power series represenitation of $$1 \over {(1-x)^3}$$ which is valid in the interval $(-1,1)$. Is it right to use the product of $1 \over {1-x}$ and $1 \over ...
7
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1answer
242 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 $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i}$ ...
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0answers
2k views

power series of arcsin(x) centered at x = 0

I am trying to prove that the Taylor expansion of $\arcsin(x) = \sum\limits_{n=0}^\infty \cfrac{(2n!)x^{2n+1}}{(2^nn!)^2(2n+1)}$. Sorry about the notation, I'm not sure what syntax to use. S stands ...
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4answers
139 views

Absolute convergence of the series $\sum_{n=1}^{\infty} (-1)^n \ln\left(\cos \left( \frac{1}{n} \right)\right)$

This sum $$\sum_{n=1}^{\infty} (-1)^n \ln\left(\cos \left( \frac{1}{n} \right)\right)$$ apparently converges absolutely, but I'm having trouble understanding how so. First of all, doesn't it ...
2
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1answer
160 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 ...
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1answer
45 views

Trying to show that the product of two power series equals 1.

I've reduced a large homework problem to the following smaller problem. Let $P = \sum_{i=0}^\infty a_i X^i$ denote a formal power series over a field. Assume $a_0 \neq 0$, and define $Q = ...
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2answers
75 views

If $f$ is the sum of a convergent power series on a disk $D(0;R)$ prove that the integral of $f$ over any closed path $\gamma$ in $D(0;R)$ is zero

If $f$ is the sum of a convergent power series on a disk $D(0;R)$ prove that the integral of $f$ over any closed path $\gamma$ in $D(0;R)$ is zero. How can I able to prove the above problem without ...
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1answer
166 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
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1answer
364 views

Adding two Power series or Maclaurin sums together and their radius of convergence

Say you have two power series. One of them has ROC of 2, and the other one has an ROC of 4. If you add the two series together is the ROC ALWAYS the lesser ROC? It seems to be a trend I've noticed, ...
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0answers
87 views

Taylors Inequality to evaluate $f(x) = x\sin(x)$ when $a = 0$ and $-1\le x\le1$

Trying to calculate the error of this function when you use a Taylor expansion to degree 4. I keep getting $.039$ when the answer in the back of the book is $.042$. I take the fifth derivative of ...
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0answers
80 views

Expansion in powers

Let $n=2k, k \in Z_+$. Let $$P_k\left(\frac{t}{\sqrt n}\right)=n!\sum_{\begin{smallmatrix} n_1+\ldots+n_k=n \\ ...
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3answers
75 views

Is the degree function well behaved over power series?

For non-zero formal polynomials $x$ and $y$ it holds that $\deg(xy)=\deg x + \deg y$. Allowing for infinite degrees, does this formula hold for arbitrary non-zero power series? And is there a ...
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1answer
49 views

Notation for the coefficient of the $i$th term of formal power series.

What notation is standard for the coefficient of $X^i$ in a formal power series $P$? I was thinking of $X^i \cdot P$, by analogy with the dot product.
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2answers
49 views

Lagrange remainder to approximate $3^{2.1}$ less than 0.1

How do I solve this problem: Use the appropriate Taylor polynomial $P_n(x,c)$ to estimate $3^{2.1}$ with error less than $0.1$, given $\ln 3$ is about $1.099$. I understand that the remainder ...
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1answer
55 views

How do I show that these sums are the same?

My textbook says that I should check that $$ \sum_{i=0}^\infty \frac{\left( \lambda\mathtt{I} + \mathtt{J}_k \right)^i}{i!} $$ is in fact the same as the product of sums $$ \left( \sum_{i=0}^\infty ...
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1answer
197 views

Find taylor polynomial that approximates e^x with accuracy at least 1.

Find Taylor polynomial at $x=0$ which approximates $e^x$ with accuracy at least $1$ for each $x \in [-2,2]$. I dont undestand these questions that involve the $n^{th}$ remainder. I know I need to ...
2
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2answers
4k views

Maclaurin Series for $\arctan(x)$ by successive differentiation

I am trying to find a Maclaurin Series for $\arctan(x)$ up to the term with the fifth power of x and I have to use the method of successive differentiation. I know (from an example in my notes) the ...
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2answers
55 views

Radius of convergence in a series. Ratio test.

I am having a hard time with this question. $$\sum_{k=0}^{\infty} \frac{-(1)^k (4^k -3)x^{2k}}{k^4+3}$$ I used the ratio test and got stuck here: $$x^2 \lim_{k\to\infty} \frac ...
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2answers
4k views

Power series for $\ln(1+x^2)$

In the problem I am asked to use a power series representation of $\ln(1+x)$ to approximate the integral from $0$ to $0.5$ of $\ln(1+x^2)$ to within 4 decimal places. So far I have found a series for ...
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1answer
202 views

Radius of convergence of $\sum_{-\infty}^{\infty}3^{-|n|}z^{2n}, z \in \mathcal{C}$

I want to find out the radius of the following power series of a complex variable: $\sum_{-\infty}^{\infty} 3^{-|n|} z^{2n}, z \in \mathbb{C}$ The ration test $\lim_{n \to ...
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1answer
113 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 ...
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1answer
53 views

Intuition Behind Krantz Theorem

The theorem I'm referring to is as follows: Let $z_0$ be a root of a nonzero holomorphic function $f$ , and let $n$ be the least positive integer such that, the $n$-th derivative of $f$ evaluated ...
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0answers
98 views

Relating terms in differential equation with power series

Having problems with a task on a differential equation containing a power series. Given $$\frac{dx}{dt} = \lambda x + \sum_{n=2}^\infty b_n x^n$$ $$\frac{dy}{dt} = \lambda y$$ $$x(y) = y + ...
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2answers
921 views

Using Generating Functions to Solve Recursions

I have the recursion $A(n) = A(n-1) + n^2 - n$ with initial conditions $A(0) = 1$. I attempted to solve it using generating functions and I'm not quite sure I have it right, so I thought I might ask ...
3
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2answers
104 views

Linear Algebra Matrix Question

I am having trouble showing that $e^AX = Xe^A$ for all $n$ by $n$ matrices $X$ where $A$ is an invertible $n$ by $n$ matrix iff $AX = XA$ for all $X$. Any help will be appreciated. Thank you
2
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2answers
49 views

Approaching Maclaurin/Taylor series from a different angle.

So I have been asked a question for one of my problem sets but I haven't found a "good" way to approach it. Personally I was thinking of approaching the series it backwards but then again if I want to ...
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1answer
114 views

A question regarding representation of a function as a power series

I'm trying to help my brother with a calculus problem related to the representation of a function as a power series. The task is to find what power series is represented by the following function, and ...
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2answers
1k views

Power series by partial fractions

I am trying to find a power series centered at the origin for the function $f(z) = \frac {1}{1-z-2z^2}$ by first using partial fractions to express $f(z)$ as a sum of two simple rational functions. If ...
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3answers
513 views

Index of summation shift

I'm learning about power series in Differential Equations. Right now I'm learning about shifting summations and something that is bothering me is the following: Take the equation $$F(x) = (x-3)y' + ...
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2answers
273 views

How to construct this Laurent series?

How do I construct the following Laurent series (clipped off Wolfram Alpha)? I know that the numerator can be written as $-1+\frac{\pi}2 z-...$ Alternatively (without the Laurent series), how can I ...
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0answers
163 views

Power Series and Matrices

I am trying to prove that if a function $f(x)$ can be written as a power series in the form \begin{equation} f(x)=\sum_{n=0}^{\infty}c_n(x-x_0)^n \end{equation} such that $|x-x_0|<r$, then ...
4
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2answers
259 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 ...
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5answers
208 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, ...
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2answers
68 views

Determine if series are convergent in $(C([0,1]),\| \cdot\|_{\infty})$

Determine whether the following series $\sum_{n=1}^\infty f_n$ are convergent in the space $(C([0,1]),\|\cdot\|_\infty)$, where (i) $f_n(t)= \frac{t^n}{n!}$; (ii) $f_n(t)=\frac{t^n}{2n}$ ...
1
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1answer
222 views

using power series expansion to find a holomorphic function which solves a differential equation

Using power series expansions, find a function $f$ which is holomorphic on the unit disk $D:=$ {$z\in\mathbb C:|z|<1$} and solves the differential equation $(1-z^2)f''(z)-4zf'(z)-2f(z)=0$ for ...
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1answer
361 views

Evaluate a complex integral using power series expansions

Using power series expansions, evaluate the integral $$\int_{\gamma_r}\sin\left(\frac{1}{z}\right)dz.$$ where $\gamma_r:[0,2\pi]\rightarrow \mathbb C$ is given by $\gamma_r(t)=r(\cos t + i\sin ...
2
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2answers
726 views

Confused by Laurent series

A typical problem related to Laurent series is this: For the function $\frac 1{(z-1)(z-2)}$, find the Laurent series expansion in the following regions: $\\(a) |z|<1, \\ (b) 1<|z|<2, ...
2
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1answer
282 views

Confused over analytic functions, point convergence of power series

It is well-known that a power series sums to a function that is analytic at every point inside its circle of convergence and that conversely, if a function is analytic on an open disc then its Taylor ...
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2answers
165 views

How do I obtain the Laurent series for $f(z)=\frac 1{\cos(z^4)-1}$ about $0$?

I know that $$\cos(z^4)-1=-\frac{z^8}{2!}+\frac{z^{16}}{4!}+...$$ but how do I take the reciprocal of this series (please do not use little-o notation)? Or are there better methods to obtain the ...
2
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2answers
290 views

Solving limit by substituting a power series

I dont understand why I am getting 2 and the textbook says it is -2. $$\lim_{x\to 0} \frac{1-e^x}{\sqrt{1+x}-1}$$ I subbed the power series for $e^x$ and $(1+x)^{1/2}$ then got rid of the $1$ on top ...
13
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1answer
331 views

counterexample to Abel's theorem

$D=\{z\in\Bbb C:|z|<1\}$. Let $f(z)=\sum\limits_{n=0}^\infty a_n z^n(a_n,z\in\Bbb C)$ be a power series, the radius of convergence of $f$ is $1$, $\sum\limits_{n=0}^\infty a_n =s$. Give a ...
2
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1answer
114 views

Coefficients of Generating Functions

This problem is from Stanley's Enumerative Combinatorics: Volume 1. page 115 here for those desirous of context (prettier conTeXt). Anyway, it asks for fixed $j,k\in \mathbb{Z}$ to show that ...
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3answers
125 views

What is the significance of this theorem (coefficients of power series as integrals)?

Isn't it easier to obtain the series' coefficicients by differentiation rather than by integration? The above text uses this theorem as an intermediate step in obtaining the generalised Cauchy ...
0
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1answer
107 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 ...
1
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3answers
46 views

Rational polynomial from coefficents

Given two polynomials $$ p(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_{n-1}x^{n-1} \\ q(x) = b_0 + b_1 x + b_2 x^2 + \ldots + b_{n}x^{n} $$ And the series expansion from their rational polynomial $$ ...
12
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2answers
466 views

Proving that $\pi=\sum\limits_{k=0}^{\infty}(-1)^{k}\left(\frac{2^{2k+1}+(-1)^{k}}{(4k+1)2^{4k}}+ \frac{2^{2k+2}+(-1)^{k+1}}{(4k+3)2^{4k+2}}\right)$

Long time ago I've been playng with formulas for $\pi$ and found that one above in the title which can also be expressed as \begin{align*} ...