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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79 views

What is the sum of the series $\sum_{n=1}^{\infty}\frac{sin(\frac 1 n)}{n}?$

I want to explicitly calculate the sum of the following series $$\sum_{n=1}^{\infty}\frac{sin(\frac 1 n)}{n}$$ I know that the series converges absolutely because of $|sin(\frac 1 n)|\leq \frac 1 ...
2
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1answer
37 views

Prove that the series $\sum_{1}^{\infty}a^n \frac{logn}{n^2}$ is convergent only for $[-1,1]$ and divergent elsewhere.

Prove that the series $\sum_{1}^{\infty}a^n \frac{logn}{n^2}$ is convergent only for $a \epsilon [-1,1]$and divergent elsewhere. I have given it an honest attempt, I can see why this must be true ...
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2answers
27 views

Determining radius of convergence by factoring

I know that you can determine radius of convergence of a function $f(x)$ by factoring the function to look like $1/(1-x)$. But when I tried to do that here, it didn't work. This is my work: But the ...
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2answers
33 views

How to apply Abel's theorem

In this example, where does the term $log(1+x)$ come from? And can someone briefly summarize the steps we take when we want to apply Abel's theorem? Thanks!
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2answers
26 views

Find the N coefficient of a expansion

I have given expression $(X+X^a+X^b+X^c+X^d+....)^k$ where $a,b,c,\ldots$ and $k$ are natural numbers. How we can find the $Z$ coefficient, i.e, $X^Z$ of the above term $1\le Z \le k$?
2
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2answers
42 views

How to deal with power series that has $x^{2n+1}$, instead of $x^{n}$?

I'm wanting to find the radius of convergence and interval of convergence for a power series that isn't in the form $$\sum_{n=1}^{\infty} a_{n}x^{n}\tag{1}$$ but is instead in the form $$\sum_{n=1}^{\...
3
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0answers
39 views

Taylor series Lagrange Remainder explanation

So, given a Taylor series: $$f(x)=f(x_0)+f'(x_0)(x-x_0)+f''(x_0)\frac{(x-x_0)^2}{2!}+\cdot\cdot\cdot+f^{(n)}(x_0)\frac{(x-x_0)^n}{n!}+R_n$$ The error $R_n$ is given by: $$R_n=\frac{f^{(n+1)}(\xi)}{(n+...
2
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1answer
21 views

Manipulating complex power series, finding terms

In an exercise I am being asked to find the terms up to order $\leq 3$ in the power series of $$a)\frac{1}{\cos(z)}$$ $$b) \frac{\sin(z)}{\cos(z)}$$ By definition, $\sin(z)=\frac{e^{iz}-e^{-iz}}{2i}$ ...
0
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0answers
19 views

Radius of convergence of two particular power series is the same

I am trying to show the following: Let $m \in \mathbb N$ be fixed, show that the sets of convergence of the series $\sum_{n=1}^{\infty} a_nz^n$ and $\sum_{n=1}^{\infty} a_{n+m}z^n$ are the same. ...
2
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1answer
41 views

Determine behaviour of $\sum_{n=1}^{\infty} \frac{1}{n}z^n$ for $|z|=1$

I am trying to analyze behaviour of the series $\sum_{n=1}^{\infty}\frac{1}{n}z^n$ at the set $\{|z| \leq R\}$ where $R$ is the radius of convergence of the series. Since $\lim_{n \to \infty} \frac{...
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2answers
93 views

Identifying the series $\sum\limits_{k=-\infty}^{\infty} 2^k x^{2^k}$

I came across following bi-infinite sum: $\sum_{k=-\infty}^{\infty} 2^k x^{2^k}$ Is this a known series? After some plotting I have the feeling that it could be equal or very similar to $-\frac{1}{...
1
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2answers
28 views

Solve the recurrence $a_n=3a_{n/3}+2$ given $a_0=1$ and $n$ is a power of $3$

Solve the recurrence $$a_n=3a_{n/3}+2$$ given $a_0=1$ and $n$ is a power of $3$ I am trying to study for my final using my previous quizzes, of which I got this question wrong. My instructor wants me ...
3
votes
1answer
44 views

Radius of convergence of two power series

I am trying to find the radius of convergence and trying to figure out the behaviour on the frontier of the disk of convergence of the following power series: a) $\sum_{n=1}^{\infty} \dfrac{n!}{(2-i)...
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0answers
33 views

Find an analytic continuation

Let $f(z)=\sum_{j=0}^{\infty}z^j$ for $|z|<1$. For what values of $\alpha$ ($|\alpha|<1$) does the Taylor expansion of f(z) about $z=\alpha$ yield a direct analytic continuaton of f(z) to a disk ...
1
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1answer
44 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. ...
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0answers
9 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 $f(x)$ ...
1
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3answers
42 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
20 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 $...
0
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1answer
40 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 ...
2
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2answers
28 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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1answer
25 views

Developing in power series [closed]

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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3answers
44 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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2answers
47 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
24 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 ...
1
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1answer
28 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 $$\left(\frac{1}{1+t}\right)^a\left(\frac{1}{1-...
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0answers
25 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 $10^{-2}.$ ...
0
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1answer
22 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.
2
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1answer
36 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) ...
13
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2answers
339 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 $1728=12^3$....
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0answers
32 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 ...
13
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4answers
503 views

How can we show that $ \sum_{n=0}^{\infty}\frac{2^nn[n(\pi^3+1)+\pi^2](n^2+n-1)}{(2n+1)(2n+3){2n \choose n}}=1+\pi+\pi^2+\pi^3+\pi^4 ?$

We proposed this sum, but we are lacking in knowledge of this area of maths and we would ask if any of the authors would be willing to show us step by step how to go about proving this sum. $$ \sum_{n=...
2
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1answer
34 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}= 1+r+r^4+r^{9}+...
0
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0answers
27 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
votes
1answer
72 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 \mathrm{d}x\...
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0answers
57 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 ...
1
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1answer
62 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!} \sum_{k=0}^{\...
2
votes
4answers
138 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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2answers
37 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 $\pi/2$). $\left\|\...
0
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2answers
51 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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4answers
71 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 $x_0=...
0
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1answer
48 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
votes
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
votes
3answers
128 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
vote
1answer
35 views

What is the Laurent expansion of $f(z)=\frac1{z-3}$?

What is the Laurent expansion of $f(z)=\dfrac1{z-3}$? In the region, $|z-3|>0$ ? I just computed the Laurent expansion in the region $|z|>3$ by dividing the denominator by $\dfrac1z$ and ...
1
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5answers
56 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 ...
0
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0answers
22 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
votes
1answer
43 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)  &...
0
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2answers
44 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 $y=\sum_{n ...
1
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1answer
36 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. $$r^{2}a_{0}x^{r}+(r^{2}+2r+1)a_{1}x^{r+1}+\sum^{\infty}_{0}\big[[(n+r+2)(n+r+1)+(...