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
51 views

Convergence of a series that looks similar to $e^x$

Suppose I have some $\epsilon > 0$ and some constant $c > 0$. Does the series $$ \sum_{n=1}^{\infty} \frac{c^{n^{\epsilon}} }{[n^{\epsilon}]!}, $$ where $[r]$ is the integral part of a real ...
5
votes
3answers
293 views

$\sin^2(x)+\cos^2(x) = 1$ using power series

In an example I had to prove that $\sin^2(x)+\cos^2(x)=1$ which is fairly easy using the unit circle. My teacher then asked me to show the same thing using the following power ...
0
votes
2answers
60 views

Retrieving $f(x)$ from Taylor Series

I've been trying to extract the $f(x)$ from the following Taylor series: $$ \sum_{k=1}^{\infty} (-1)^{k}\frac{kx^{k+1}}{3^{k}} $$ I moved things around a bit to make it look like this: $$ ...
0
votes
1answer
44 views

Find the radius of convergence and interval of covergence of the series.

I'm in the process of trying to learn about the topic of series, power series, etc. (when I say learn I mean self-teach... everything urgh). My general understanding of the topic of Series as a whole ...
1
vote
0answers
58 views

Calculus(power series)

Hi could anyone help me solve this problem.(without expanding the function). I cant seem to get it. Express the function as a power series $$\ln\left(\frac{1+x}{1-x}\right)$$ I managed to split the ...
0
votes
3answers
62 views

Finding interval of convergence for series

Find the interval of convergence and radius of convergence for the series: $$ \sum_{n= 0}^\infty \frac{x^n}{3^n} $$ I'm not sure if I'm correct, but would the interval of convergence be $(-3,3)$ ...
3
votes
1answer
138 views

How to estimate the growth of a “savage” function near 1?

Say I have a function which exists within the unit disk, say $$f(x)=a_0+a_1x+a_2x^2+...$$ If we know sufficient information about the coefficients, say we know the growth rate of ...
1
vote
1answer
61 views

Proving that the derivative of the integral of a power series equals the original power series

I've been thinking about the following recently: If we have a power series $f(x) = \sum_{n=0}^\infty a_n(x-c)^n$ and $F(x)=\sum_{n=0}^\infty \frac{a_n}{n+1}(x-c)^{n+1}$ where $F(x)$ is constructed by ...
3
votes
2answers
76 views

Finding terms of a Taylor series where $f(x)$ is a function with a power

I've been stuck with this Taylor series problem for a while now. We have that $$ f(x) = (1 + x^2)^{-2/3} $$ and it's centered at $0$. So what I thought of doing was the $$ \frac{f^{n}(a)(x - ...
2
votes
1answer
99 views

Radius of Convergence of product of power series

Is the following statement true? If $P(z)$ is a power series over $\mathbb C$, then $ P(z) $and $P(z)^n$ have same radius of convergence for any positive integer n.
1
vote
3answers
38 views

A question about sum of probability… if $P(X\ge n)-P(X=n)=P(X>n)$

If we know that $P(X\ge n)=(1-p)^{n-2}$ (This is not the main subject of the question, so I wont explain about it, hope this OK, but in sort: we get it because $P(X\ge n)=\sum_{k=n}^\infty ...
0
votes
1answer
33 views

What is the sum of this series: $\sum_{k=n}^{\infty}(1-p)^{k-2}\cdot p$?

I have: $$\sum_{k=n}^{\infty}(1-p)^{k-2}\cdot p$$ $|p|<1$. The sum is: $$=\frac{(1-p)^{n-2}\cdot p}{1-(1-p)}=(1-p)^{n-2}$$ Or I wrong? I use the fact that the sum of a series is: $\frac{a}{1-p}$ ...
0
votes
1answer
293 views

Complex Power series with factorials

Find the radius of convergence of $$\displaystyle\sum_{n=0}^\infty z^{n!}$$ $$\displaystyle\sum_{n=1}^\infty {(-1)^nz^{n(n+1)}}/{n}$$ What is the behavior of the series for $z=1, -1,i$
1
vote
3answers
297 views

Find complicated Taylor Series

According to some software, the power series of the expression, $$\frac{1}{2} \sqrt{-1+\sqrt{1+8 x}}$$ around $x=0$ is $$\sqrt{x}-x^{3/2}+\mathcal{O}(x^{5/2}).$$ When I try to do it I find that I ...
5
votes
2answers
91 views

Closed form of generating function consisting of power of two binomials

Let $g(x)$ be infinite formal power series and $$g(x) = (1 + x)(1 + x^2)\cdots(1 + x^{2^k})\cdots$$ Show that $(1 - x) g(x) = 1$. My book gives following proof: Using a fact that $(1 - x^k)(1 + ...
0
votes
2answers
86 views

How do I find the coefficients in this product?

$(x+x^2+x^3+x^4+x^5+x^6)^5$ How do I find the coefficients of this product quickly?
0
votes
1answer
157 views

How to Expand $f(x) = \sqrt{x^2-y^2}$ about $x=y$

How would one go about expanding $f(x) = \sqrt{x^2-y^2}$ about $x=y$? You can't do a Taylor series, because $f'(x=y)$ diverges. I tried a few other things, but to no avail. Interestingly enough, ...
0
votes
2answers
63 views

Convergence of complex power series question

I need some help to solve this problem and find the domain of convergence of the following power series: $$\displaystyle\sum_{n=0}^\infty(2^n+i^n)(z-2i)^n$$ Thank you!
0
votes
1answer
37 views

Radius of convergence problem

I need a hint on this problem, been staring on the blackbord a long time now. Problem: Suppose $f(x) = \sum a_{n}(x-x_{0})^n$ has radius of convergence $R$ and $0 < r < R_{1} < R$. Show that ...
1
vote
2answers
53 views

polynomial series and root multiplicity

Excuse me, because I know this is a double post but I can't for the life of me find the original post. Given a sequence $(a_n)$, one can construct a polynomial of the form ...
3
votes
1answer
115 views

Sum of combinations series

What is the value or tight upper limit of the following summation: $$\sum_{k=0}^n{n\choose k} x^{k(n-k)}$$
0
votes
2answers
132 views

Limits of nth-root on various sequences

I have been computing the ratius of convergence for different power series using the the root-test. In many of them, I am left with nth-root expressions that computationally approaches 1 when n goes ...
1
vote
2answers
80 views

Power Series Representation…

I am having a hard time understanding how to proceed with this question... Find a power series representation for the function and determine it's radius of convergence $$ f(x)= x^2\ln(1+x^2) $$ How ...
1
vote
2answers
94 views

Simplifying ratio test with exponents $k+1$

Question: Find the interval and radius of convergence. $$\sum_{k=1}^\infty\frac{(x-1)^k(k^k)}{(k+1)^k} .$$ I applied the ratio test. ...
3
votes
1answer
54 views

“Reduction of Dirichlet series into power series”

In a paper of Riemann, he states to following formal identity. If $f(s)=\sum\limits_{k=1}^{\infty}\frac{a_k}{k^s}$ and $F(x)=\sum\limits_{k=1}^{\infty}a_kx^k$ then ...
1
vote
1answer
200 views

Solve the functional equation $\,\,f(2x)=2x f^\prime(x)$

If $f(x)$ is a real analytic functions on $\mathbb R$, and $$2xf'(x)=f(2x),$$ then find $f(x)$. My idea: express $f$ as: $$f(x)=\sum_{n=0}^{\infty}a_{n}x^n.$$ Thank you
0
votes
1answer
57 views

Close form of a power series starting at $n=2$

This is the power series I am looking at $\sum_{n=2}^{\infty}{n(n-1)z^n}$. I want to find the closed form of this power series. This is my approach, if I divide the power series by $z^2$, then I ...
4
votes
3answers
721 views

Find a closed form from the given power series

I have the power series $\sum_{n=0}^{\infty} {z^{2n}\over{n!}}$, how do I find the closed form for this power series. I am aware that $e^z=\sum_{n=0}^{\infty} {z^{n}\over{n!}}$, so I tried to ...
1
vote
1answer
29 views

Converting from radius of convergence to interval of convergence

Using the root test I have determined that $$\sum n^{-n} x^n$$ has a radius of convergence of infinity and $$\sum n^{n} x^n$$ has a radius of convergence of 0. Does this mean that the respective ...
1
vote
1answer
180 views

Showing the power series of $\cos(x)$ converges uniformly to $\cos(x)$ on every bounded interval

Show that the power series of $\cos(x)$ converges uniformly to $\cos(x)$ on every bounded interval. My attempt: The power series for $\cos(x)$ is $$\sum_{n=0}^{\infty} ...
3
votes
1answer
36 views

Power series with interval of convergence of $(-1,1]$?

Is there a power series with an interval of convergence of $(-1,1]$? Wouldn't the fact that absolute convergence implies regular convergence make such a function impossible to find?
2
votes
4answers
373 views

Why is my series wrong?

Why is this series wrong and how does it differ from this other one? We had to find the general term for the series: $ 1/3+2/9+1/27+2/81+1/243+2/729+\ldots $ where the index begins at $n=1$ So I came ...
1
vote
1answer
67 views

Maclaurin Series application

This was something that popped up in class and confused me... So knowing that the power series may be differentiated term by term inside the interval of convergence, using the Maclaurin series you ...
0
votes
1answer
84 views

Does a series always diverge if its sequence isn't a null-sequence?

I have the following series: $$\sum_{n = 1}^\infty 2^{n^2}z^n$$ The task is to give its radius of convergence. I solved that one using the root-test and came to the same answer. But the solution ...
0
votes
2answers
239 views

Finding coefficients of laurent series for $\frac 1{1-\cos z}$ about zero

I know I'm wrong, but I fail to see why I'm wrong. My goal is to try and find the terms for the Laurent series of $f(z)=\frac{1}{1-\cos(z)}$ but I'm surely off. $$\begin{align} f(z)&= ...
0
votes
1answer
133 views

Multiplicative Inverse of a Power Series

For a formal power series $$F(x) = \sum p_i x^i$$ a multiplicative inverse of $F$ exists iff $p_0 \neq 0$. The inverse $\sum q_i x^i$ satisfies the recursion $$q_0 =\frac{1}{p_0}\\ q_{n} = ...
2
votes
0answers
39 views

Summing the series $\sum_{k = -\infty}^{\infty} e^{ja_{0}^kt}$

For a fixed $t$, I wish to find the value of, $S = \sum_{k = -\infty}^{\infty} e^{ja_{0}^kt}$ where it is known that $a_{0} > 1$. I tried to express $S$ as a $z$ transform of some known function, ...
0
votes
1answer
92 views

Find the radius of convergence of the power series

$\displaystyle\sum_{n=0}^{\infty}a_nz^n$, where $a_{2k+1} = 2^k$ and $a_{2k} = (1 + (1/k))^2$ for $k = 0, 1, 2, \dotsc$ I started off by doing the ratio test, but I know that the ration test is for ...
0
votes
3answers
61 views

How do I simplify the formal power series $1+2x^2F(x)^2$?

I have this formal power series $$F(x)=1+2x^2F(x)^2$$ that I want to put into non-recursive form. I can expand, $$1+2x^2F(x)^2=1+2x^2(1+2x^2F(x)^2)^2= 1+2x^2+8x^4F(x)^2+8x^6F(x)^6$$ and I could ...
2
votes
1answer
38 views

Power series centered at $x =0$

I have this question in my advanced calculus textbook. Give an example of a power series centered at $x = 0$ that converges on $[-2,2]$ but not absolutely on the entire interval $[-2,2]$, and ...
1
vote
0answers
17 views

When is a series expansion related to its derivative by a polynomial equation?

Is there some common theory behind the following two examples? Example 1. Let $p(t) = \sum_{n \geq 0} (-1)^k t^{2k}/(2k)!$, and $x = p(t), y = p'(t)$. Then $x^2 + y^2 = 1$ identically. Example 2. ...
2
votes
0answers
25 views

Finding a base for a series to sum to a constant

I'd like to find the value of $r$ that solves the following equation: $$\sum_{n=1}^N r^{\frac{-1}{n}} = C \,,$$ where $N$ and $C$ are positive constants. An approximate method would also work fine ...
8
votes
2answers
405 views

Taylor Series of $\frac{1}{1-\cos x}$

The problem is, as the title suggests, to find the Power Series Expansion of $\frac{1}{1- \cos x}$ around $x=c$. What I've tried: Direct Computation: Derivatives get very ugly quickly, and don't ...
0
votes
2answers
42 views

Question on generating power series for a function

When generating power series for the function $y = 1/(2-x)$ I can see two different ways of solving this question, but with very different answers. SOLUTION: $$y=\frac{1}{1-(x-1)} = ...
1
vote
1answer
79 views

Radius and Interval of Convergence for Power Series

Find the radius and interval of convergence for the power series $\displaystyle{\sum_{k=1}^{\infty}} \frac{(x+3)^k}{k(6+(-1)^k)^k}$ I found that R=1 by calculating $\frac{1}{R} = ...
2
votes
0answers
377 views

What does the convergence of a Dirichlet series tells us about the convergence of a power series?

If $D(s)=\displaystyle \sum_{k\geqslant 1} f(k)\, k^{s}$ converges for $\Re(s)\lt a$, what is the radius of convergence of $\displaystyle \sum_{k\geqslant 1}f(k)\, x^k$ $=T(x)$? Conversely, what ...
0
votes
1answer
50 views

Term wise differention

Consider $S(x) = \displaystyle{\sum_{k=1}^{\infty}} x^k k^2$. (a) Find an explicit formula for $S(x)$ on the interval $-1<x<1$ by repeated termwise differentiation of a geometric series. Be ...
1
vote
1answer
32 views

Finding the Power Series of a Complex fuction.

Find a power series expression $\sum_{n=0}^\infty A_n z^n $ for $ \frac{1}{z^2-\sqrt2 z +2} $ I'm completely stuck on this question. I know how to manipulate power series but I've never had to find ...
9
votes
4answers
239 views

Closed form of $\sum \frac{x^n}{n^n}$

Is there a closed form of this series? $$ f(x) = \sum_{n=1}^\infty \frac{x^n}{n^n} $$ I tried few standard tricks how to sum a power series but none of them helped.
1
vote
1answer
91 views

Solving differential equations using power series

I need to solve this differential equation by power series: $$y''+3xy'+(2x^{2}+6)y=0$$ Any help is great! Thanks!