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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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
votes
0answers
25 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 ...
-1
votes
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
70 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\...
1
vote
0answers
56 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
vote
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
135 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 \...
0
votes
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
votes
2answers
48 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
votes
4answers
69 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
votes
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
32 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
vote
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
votes
0answers
21 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
votes
2answers
43 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
vote
1answer
35 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)+(...
2
votes
2answers
58 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
34 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
votes
1answer
20 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
vote
1answer
37 views

Proving that a function is real-analytic

I try to solve the following exercise: Let $f:\mathbb{R}\to\mathbb{R}$ with $f(x):=\frac{1}{1+x^4}$. Prove that $f(x)$ is real analytic and compute the radius of convergence of it's Taylor series at ...
0
votes
2answers
44 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 $f(x)...
1
vote
1answer
37 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.
3
votes
2answers
249 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 ...
0
votes
1answer
39 views

Power series solution to $y' = y(1-y)$

Find the first five terms of the power series solution to the differential equation: $$y' = y(1-y)$$ Letting $y = a_0+a_1x+a_2x^2+a_3x^3+...$ It's evident that: $$y' = \frac{dy}{dx} = a_1+2a_2x+...
0
votes
1answer
50 views

Show that $e^{\varepsilon |x|^{\varepsilon}}$ grows faster than $\sum_{k=0}^{\infty} {|x|^{2k}}/{(k!)^2}$

I am wondering whether we have for $$f(x):=\sum_{k=0}^{\infty} \frac{|x|^{2k}}{(k!)^2} $$ that $$\lim_{x \rightarrow \infty} \frac{e^{\varepsilon |x|^{\varepsilon}}}{f(x)} = \infty$$ for any $\...
0
votes
1answer
56 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 ...
0
votes
1answer
30 views

Expansion of $f = \sum_{n=1}^\infty \frac{1}{2^n} \frac{z^n}{1 - z^n}$ in power series around $z = 0$

Let $f = \sum_{n=1}^\infty \frac{1}{2^n} \frac{z^n}{1 - z^n}$, for $z \in \mathbb C \setminus ${$z \in \mathbb C: \exists n \geq 1,\quad z^n = 1$}. By the ratio test, the series converges when $|z|&...
2
votes
1answer
39 views

Consider the power series $\sum_{n=1}^\infty(-1)^{n-1}\frac{x^{2n+1}}{(2n+1)(2n-1)}$.

Consider the power series $\sum_{n=1}^\infty(-1)^{n-1}\frac{x^{2n+1}}{(2n+1)(2n-1)}$. Find a closed form expression for all x which converge and hence evaluate $\sum_{n=1}^\infty\frac{(-1)^{n-1}}{(2n+...
1
vote
2answers
18 views

Why is it that the interval of convergence is half open?

I am given the following power series and asked to find the radius of convergence and determine the exact interval of convergence $$\sum\biggr(\frac{3^n}{n\cdot 4^{n}}\bigg)x^n \Leftrightarrow \sum\...
1
vote
2answers
24 views

What is the power series and domain for this function?

$$f(x)= \frac{x}{1+5x^2}$$ I got the power series: $$\sum_{n=0}^\infty (-1)^n (5^n)(x^{2n+1})$$ Assuming this is correct I would think the domain would be $$(-5^{1/3}, 5^{1/3})$$ because the absolute ...
1
vote
2answers
52 views

Show that a power series is analytic inside its radius of convergence

Let $f(z)=\sum_{k=0}^\infty a_k(z-z_0)^k$ with radius of convergence $R$ then $f$ is analytic on the open disk around $z_0$ with radius $R$. What I was thinking about is an approach based on this ...
0
votes
0answers
21 views

Continuity of series implies continuity of coefficients?

For each $t\in [0,1]$ let $f_t(z)$ be an entire function. By holomorphicity it equals its own Taylor series: $$f_t(z) = \sum_{n=0}^\infty a_n(t)\,z^n, \qquad \forall \,z\in\mathbb{C}.$$ Now ...
0
votes
0answers
29 views

Solving a power series centered at 0 by integrating another power series

Problem: Find the power series for g(x) centered at 0 by integrating the power series f(x). Give answer in sigma notation along with the first four nonzero terms. $$g(x)=\ln{(1-3x)}\space \land \...
0
votes
1answer
32 views

Power series expansion of a complex function

How can I depict function $f(z)=\sqrt{1+\sqrt{1+z^2}}$ as a power series around zero? Where Log is a function going from $C \setminus (-\infty,0]$? Since with such a logarythm $Re(\sqrt{z}>0$ then ...
5
votes
2answers
58 views

How to evaluate $\lim\limits_{x\to 0}\frac{e^{\arctan{(x)}}-xe^{\pi x}-1}{(\ln{(1+x)})^2}$?

How to evaluate $\lim\limits_{x\to 0}\frac{e^{\arctan{(x)}}-xe^{\pi x}-1}{(\ln{(1+x)})^2}$? So I think we expand to $x^2$ since the lowest term for $\ln(1+x)$ is $x$ Let $u=\arctan{(x)}$ $\lim\...
0
votes
1answer
10 views

How to compute taylor series of $\sin{x}$ about $a=\frac{\pi}{4}$?

How to compute taylor series of $\sin{x}$ about $a=\frac{\pi}{4}$? We know $\sin{x}=\sum^{\infty}_{n=0}\frac{(-1)^nx^{2n+1}}{(2n+1)!}$. Let $t=x-\frac{\pi}{4}$, then $t+\frac{\pi}{4}=x$ Then $\sin{...
0
votes
1answer
38 views

Laurent Series of $(z-2)/(z+1)$ at $z=-1$ [closed]

What's the Laurent series expansion of $\frac{z-2}{z+1}$ at $z=-1$?
0
votes
2answers
82 views

Verify f'(x) = e^x

The following is a proof I wrote to prove that given $f(x)=e^x$, $f'(x)=e^x$. For this proof we must use the Taylor Series for $e^x$, $\sum\limits_{n=0}^{\infty}\dfrac{x^n}{n!}$. Since the derivative ...
0
votes
1answer
17 views

Criteria for convergence of power series

Given the power series: $\; \sum_{i=0}^{\infty}a_nz^n \;$ Proof that if there exist $s,M \in \mathbb R $ such that $|a_n|s^n \le M$ then the power series converges for every $|z|\lt s$ Can someone ...
0
votes
1answer
22 views

How do I find a power series for this function?

Given the function: $$f(x) =\frac{(11+x)}{(1-x)}$$ how would I find a power series representation? I started by rewriting the function as $$(11+x)\frac{(1)}{(1-x)}$$ and then arrived at $$(11+x)\sum_{...
0
votes
1answer
35 views

power series find values for $\sum_{n=1}^{\infty}\frac n{2^n}$ and $\sum_{n=1}^{\infty}\frac {n^2}{2^n}$

Hi I am in a basic real class and I am confused about the question: Given the geometric series: $$\frac 1{1-x}=1+x+x^2+x^3...$$ for all $|x|<1$ use results about the power series in this section ...
3
votes
2answers
54 views

Finding Exact Values of Specific Infinite Series

Prove that $\Sigma_{n=1}^{\infty}(n/2^n)=2$ and that $\Sigma_{n=1}^{\infty}(n^2/2^n)=6$. Thoughts: I have a feeling that if someone shows me how to do one, I'll be able to figure out the other. So ...
2
votes
1answer
25 views

How to compute the following series using taylor expansion manipulation?

How to compute $\sum^{\infty}_{n=0} \frac{x^n}{(n+2)n!}$ and $\sum^{\infty}_{n=0}(-1)^n \frac{(n+1)x^{2n+1}}{(2n+1)!}$ using taylor expansion manipulation? $1.\sum^{\infty}_{n=0} \frac{x^n}{(n+2)n!}=...
2
votes
3answers
55 views

Complex power series expansion of $\frac{e^z}{1+z}$

I'm trying to find complex power series expansion of $\frac{e^z}{1+z}$ centered at $z=0$ and its radius of convergence. Here is my attempt: Since $e^z = \sum_{n=0}^\infty \frac{z^n}{n!}$, we can ...
0
votes
0answers
15 views

Maclaurin polynomial expansion of $y$ about 1?

Consider the differential equation $\frac{dy}{dx}=2x+\frac{y}{x}$, where $\frac{dy}{dx}=1$ when $x=1$. Find the first three non-zero terms in the Maclaurin polynomial expansion for $y$ about $x=1$....
3
votes
2answers
45 views

Finding the power series of a complex function

So I have the function $$\frac{z^2}{(z+i)(z-i)^2}.$$ I want to determine the power series around $z=0$ of this function. I know that the power series is $\sum_{n=0}^\infty a_n(z-a)^n$, where $...
0
votes
1answer
18 views

Is a convergent power series on an open set continuous on that set?

Question in the title. If a power series $f(x)$ is pointwise (or if this is too weak, uniformly) convergent for every $x$ in an open set $U$ in the reals, is it a continuous function of $x$?