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
26 views

If the Power Series converges at x, which must be true?

I'm currently reviewing for tomorrow's Calculus BC exam but I got stuck on this one problem. $\sum_{n=0}^{\infty} a_n (x-3)^n$ converges at $x = 5$. Which of the following must be true? a. ...
2
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2answers
25 views

Finding limit of the function by power series estimation

I want to prove that the limit of function $\displaystyle \lim_{x \to \infty}\frac{\ln(x)}{x} = 0$ Of course it is easy to find it by l'hopital's rule, but i want to find it using the power series ...
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0answers
7 views

Series expansion for monotone, bounded functions

I would like to know if there is a series expansion for monotone and bounded functions, where all functions of the orthonormal basis are monotone as well. I.e. suppose that we have a function $f(x)$ ...
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3answers
31 views

Complex analysis: Using Taylor expansion to show $|c_n| ≤ \frac{1}{r^n}\sup_{z∈C_r(0)}|f(z)|$

Consider the function $f$ is defined through the power series $$f(z) := c_0 + \sum_{n=1}^\infty c_nz^n$$ and assume that the series on the right has a radius of convergence $R > 0$. Show that ...
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3answers
19 views

$f(x)=\sum_{k=0}^\infty c_kx^k$ converges for $|x|<R$ with $R>0$, $\exists x_n\ne0$ s.t. $x_n\to0$, $f(x_n)=0$ $\forall n$, then $c_k=0$ $\forall k$.

I meet a problem: The power series $f(x)=\sum_{k=0}^\infty c_kx^k$ converges for $|x|<R$ with $R>0$. If there exists a sequence $x_n\ne0$ such that $x_n\to0$ and $f(x_n)=0$ for all $n$, then ...
1
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1answer
53 views

Laurent series for $f(z) = \exp(z+\frac{1}{z})$ around $0$

I need to find the Laurent series of the following function around $0$ - $$f(z) = \exp(z+\frac{1}{z})$$ Now by power series expansion, I got $$f(z) = \sum_{m=0}^{\infty} \frac{z^m}{m!} ...
2
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4answers
112 views

Example 3.40 (b) in Baby Rudin: How to find $\lim_{n\to\infty} \sup \frac{1}{\sqrt[n]{n!}}$?

Here is Theorem 3.39 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: Given the power seires $\sum c_n z^n$, put $$\alpha = \lim_{n\to\infty}\sup\sqrt[n]{\vert c_n ...
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1answer
39 views

Integrate $\frac{\sin x^3}{x^3}$ as a power series

Today, I tried to do this by taking the MacLaurin of Sin to 4 terms, putting in $x^3$ in place of $x$, multiplying the terms by $x^{-3}$, and integrating. I came out with a sum that had $x^{6n+1}$ as ...
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0answers
31 views

Developping $\ln(x)$ in power series in $1$ [on hold]

I would like to develop $\ln(x)$ in power series in $1$. What is the way to do it around $1$ ? Thanks
2
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2answers
45 views

Determining if $A(B(x))$ is a formal power series [duplicate]

I know that by theorem, $A(B(x))$ is a formal power series if $b_0=0$. I wasn't sure if it works the other way around. Can I also say that if $b_0\ne0$, $A(B(x))$ is not a formal power series? If it ...
1
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1answer
25 views

Is this true that $\sum_{v=0}^k (-1)^v{a+v-1 \choose v}{b+k-v-1 \choose k-v}$ is the coefficient of $t^k$ in $(\frac{1}{1+t})^a(\frac{1}{1-t})^b$

I was reading a paper, in which the author assumed that $$\sum_{v=0}^k (-1)^v{a+v-1 \choose v}{b+k-v-1 \choose k-v}$$ is the coefficient of $$t^k $$ in ...
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1answer
23 views

Developing in power series

I would like to develop in power series $\frac{1}{(2+x)^3}$ and $ \frac{1}{(32-x)^{1/5}} $. What is the way to go ? Thanks
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0answers
24 views

Solving applications of power series.

The region bounded by the curves $$ y = \frac{\sin(x^2)}{x} $$ $$ y = 0, x = \frac{1}{2}$$ is rotated around the y-axis. Find the volume of the solid of revolution with accuracy to within $10^{-2}.$ ...
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3answers
40 views

What is the value of $x$ when $a^\frac{1}{x}=1$?

I used to compute complexity of an algorithm which reaches to constant value after x level because of $a^\frac{1}{x}=1$. Now I need to find $x$ to reach answer. To describe more : my recursive ...
1
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1answer
25 views

To simplify the series of matrices

Let $A$ be a square matrix in the form $A=B+O(h^2)$, where $B$ is a fixed matrix, and $O(h^2)$ is a matrix with very small elements. Assume that: $$(I-A)^{-1}=I+A+A^2+A^3+...$$ How can I esimate the ...
0
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1answer
13 views

When is the radius of convergence of the product of two complex power series twice the radius of convergence of the product of the radii

Proving that the product has a larger radius then the product isn't too bad using the nth root test, but another practice question I have asks for examples of power series $\sum a_kz^k$ with radius of ...
6
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6answers
78 views

Series of inverse function

$A(s) = \sum_{k>0}a_ks^k$ and $A(s)+A(s)^3=s$. I want calculate $a_5$. What ways to do it most efficiently?
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1answer
20 views

Finding an analytic function

enter image description here I cannot find any such function. Also, why would a function that is analytic at 0 following these criteria not be analytic on (-2,0). Thanks in advance for your help.
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2answers
173 views

Did Feynman mentally compute $\sqrt[3]{1729.03}$ by linear approximation?

In the biopic ``infinity'' about Feynman. (11:48~15:50) Feynman compute $\sqrt[3]{1729.03}$ by a mental calculation. I guess that he use the linear approximation. That is, he observe that ...
2
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1answer
33 views

Integral analog of geometric series

We all know that $$ \frac{1}{1-z}=\sum_{m=0}^\infty z^m\ , $$ for $|z|<1$. The challenge I would like to pose is: find (possibly as simple and elegant) integral representations (as many as you can) ...
0
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1answer
613 views

Taylor Series Theorem

So I see the argument presented in taylor series, that $$\sum c_n (x-a)^n = \sum \frac{f^{(n)}(a)}{n!} (x-a)^n$$ or $c_n = f^{(n)}(a)/n!$ if $x=a$ the question is, since the above only holds when ...
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0answers
22 views

The Method of Frobenius

I'm learning about the method of Frobenius and solutions about singular points. But the class didn't cover the $2^{nd}$ and the $3^{rd}$ case below. Could someone please explain the parts that I ...
0
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0answers
48 views

(solved) Holomorphy on open unit disk and continuity to the closure implies absolutely convergence of coefficients?

I am having trouble proving that the space of holomorphic functions continuous till the closure in the unit open disk coincides with the power series whose coefficients form an absolute convergence of ...
0
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4answers
63 views

Power series solution (Why the constant of the recurrence relation can be chosen arbitrarily?)

Please help me understand this: Solve $(x+1)y''-(2-x)y'+y =0$ First, since $x_0=0$ is an ordinary point, it can be guaranteed that we can find two independent power series solutions centered at ...
2
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1answer
30 views

Is there any known way to sum a subserie (square indices) of geometric series?

I was interested in the following sum. Although im not sure there exist any known way to sum this...it seems rather difficult. Can we sum for $0<r<1$ $$\sum_{k=0}^{\infty}r^{k^2}= ...
0
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0answers
22 views

Evaluation of r^3 from zero to N (Sigma Notation)

I have to evaluate the following expression, $\sum_{n = 0}^{9} (n^3 -1)$. I know that $\sum_{n = 1}^{9}(n^3 -1)$ is given by $\frac{1}{4}(9)^2(9+1)^2 - 9$ But, how do I do this from zero to ...
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1answer
22 views

Show that any polynomial of odd degree 2n+1: $f(x)=\sum_{k=0}^{2n+1} a_kx^k $, $a_{2n+1}\neq0$ has at least one real root.

Show that any polynomial of odd degree 2n+1: $$f(x)=\sum_{k=0}^{2n+1} a_kx^k $$ $a_{2n+1}\neq0$ has at least one real root. I would like to prove this using IVT, how would I go about starting ...
3
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1answer
56 views

Let $f$ be an analytic isomorphism on the unit disc $D$, find the area of $f(D)$

Let $f$ have power series $f(z) = \sum_{n=1}^\infty a_n z^n$ in $D$, then prove that $\mathrm{area}\, f(D) = \sum_{n=1}^\infty n \,|a_n|^2$. Note: We define $\mathrm{area}\, S = \iint_S ...
0
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2answers
42 views

For what values of $k$ to both of the following series converge?

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
0
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2answers
36 views

Uniform Convergence of $\sum_{i=1}^\infty \arctan\left(\frac{x}{i^2}\right)$ and its differentiabilty

I was trying to prove it is uniform convergent by it is Cauchy in sup-norm, since I don't know what does it converge to and it seems that M-test fail (as each term is bounded by $\pi/2$). ...
1
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1answer
18 views

Is this an incorrect error bound value?

In Step 3, they are determining the $(n+1)^{th}$ term. I think the proofreader just added 1, instead of subbing in (n+1). Is that right? I think the correct term should be ...
0
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1answer
43 views

Radius of Convergence of $\sum_{n = 0}^{\infty} \frac{(-1)^nn!x^n}{n^n}$

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
0
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1answer
26 views

How would I show $|x| \le 1$ given the equation for $x$ the expression in the equation?

The expression is $x = \sin(\theta /2)$. I am asking how would I show that $\sin(\theta/2)\le1$ based on the expression? I already know that the biggest $\sin$ will ever get is $[-1, 1]$ which is the ...
3
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3answers
126 views

How is the last “=” true?

How can the last equality be true? $$ G(u)=\frac{g}{(1+u)^g-1}-\frac1u=\frac{g}{gu + \cdots + u^g} - \frac{1}{u}=\sum_{k=0}^\infty c_ku^k $$
1
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1answer
15 views

What is the Laurent expansion of f(z)=1/(z-3)?

What is the Laurent expansion of f(z)=1/(z-3)? In the region, ㅣZ-3ㅣ>0 ? I just computed the Laurent expansion in the region ㅣZㅣ>3 by dividing the denominator by 1/z and making it as a geometric ...
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0answers
28 views

4th derivative of $1 - 9x + 16x^2 - 25x^3 + \dots$ [closed]

Fined the 4th derivative of $f$ at $x=0$ given that the MacLaurin series of $f$ is $f = 1 - 9x + 16x^2 - 25x^3 + \dots$.
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5answers
52 views

Does this power series $\sum_{n=0}^{\infty} \frac{n^n}{(n!)^2}x^n$ converge for all $x$?

Does this power series $\sum_{n=0}^{\infty} \frac{n^n}{(n!)^2}x^n$ converge for all $x$? It was told to me that the series does converge for all $x$, however I have investigated with a computer ...
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0answers
20 views

Determining the r value for a series solution [closed]

I have trouble answering the 2nd part of the question. For the 1st part, I just simply plugged it into the derivatives and I end up getting $r_{1}=1/2$ and $r_{2}=1/3$. For the 2nd part, I tried ...
6
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4answers
103 views

Power series solution for ODE

The ODE I have is $$y'(x)+e^{y(x)}+\frac{e^x-e^{-x}}{4}=0, \hspace{0.2cm} y(0)=0$$ I want to determine the first five terms (coefficients $a_0,\ldots, a_5$) of the power series solution ...
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0answers
18 views

Second Order Linear Non-Homogeneous DE solution with Power Series $x^2y'' - 4xy' + 6y = x^2 \cos x$

My instructor wants me to solve the above equation using power series and another method, and then to confirm the results are the same This equation does not have constant coefficients and a can't ...
0
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2answers
41 views

Solve $y'=e^{x^2}y$ (with $3$ terms only) in using power series

Solve $y'=e^{x^2}y$ (with $3$ terms only) in using power series. I know that $e^{x^2}=\sum_{n \geq0} \frac{x^{2n}}{n!}$, but I don't know how to find the coefficients $a_n$ in considering ...
0
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1answer
52 views

Taylor expansion of $\frac1{\|x-y\|}$

Let $0\neq y\in \mathbb{R^3}$ define a function $f$ on $\mathbb{R^3}$ as $$ f(x) = \frac1{\| x-y\|} $$ What are derivatives of $f$ in zero? Or equivalently, what is the Taylor series of $f$ at ...
1
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1answer
33 views

Frobenius Method to Solve a Differential Equation

Having the equation $$x^{2}y''+xy'+x^{2}y=0$$ I get the indicial equation at get r=0, and am left with the equation. ...
0
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1answer
40 views

Help with generating functions

I've got two questions. I'm trying to extract the "coefficients" of a power series. I think my terminology is incorrect here but here is what I mean. Here are some examples A(Z) = 1/(1-Z) ...
3
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2answers
166 views

Is it true that $log(i) = \frac\pi2i$ ? If so, are both of these legitimate proofs? They seem too beautiful not to be…

Sorry if this is a naive question. I have not yet taken any upper level math courses involving complex numbers. However, in preparation for those courses, together with utilizing the knowledge that ...
2
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2answers
55 views

$(x-x_0)^0$ in power series [duplicate]

When I first studied power series in high school, the teacher gave the following general definition: \begin{equation} f(x)=\sum_{n=0}^{\infty}a_n (x-x_0)^n \end{equation} He then proceeded to ...
2
votes
2answers
28 views

Difficulty finding a power series representation

I have to find a power series representation and interval of convergence for $$f(x) = \frac{x-x^2}{(1+2x)^3}$$ Noting that $\frac{1}{1+2x}=\frac{1}{1-(-2x)}=\sum_{n=0}^\infty(-2x)^n$, I start taking ...
0
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1answer
19 views

If $f(x)=\sum_{0}^{\infty}b_n(x-5)^n$ for all $x$, write a formula for $b_8$.

If $f(x)=\sum_{0}^{\infty}b_n(x-5)^n$ for all $x$, write a formula for $b_8$. Now I know that $b_n=\dfrac{f^{(n)}(5)}{n!}$. I have tried various things but I think there is something wrong with my ...
1
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1answer
34 views

How to expand the summation term with power?

How to expand the following: $$ \left( \sum^{M}_{m=0} \frac{x^{m}}{m!} \right)^{K} $$ where $M$ and $K$ are positive integers.
0
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2answers
42 views

Expand $f(x)=\log(x+ \sqrt{1+x^2})$ into power series and determine convergence intervals

We have $f(x)=\log(x+ \sqrt{1+x^2})$ and we need to expand it into power series, which I suppose is easy because $f'(x)=(1+x^2)^{-1/2}= \sum_{k=0} {-\frac{1}{2} \choose k}x^{2k}$. It follows that ...