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).

learn more… | top users | synonyms

2
votes
4answers
57 views

Compute $\frac{1^2 t}{1!}+\frac{2^2 t^2}{3!}+\frac{3^2 t^3}{5!}+\frac{4^2 t^4}{7!}+\ldots+\frac{n^2 t^n}{(2n-1)!}+\ldots$

I have to compute $$\frac{1^2 t}{1!}+\frac{2^2 t^2}{3!}+\frac{3^2 t^3}{5!}+\frac{4^2 t^4}{7!}+\ldots+\frac{n^2 t^n}{(2n-1)!}+\ldots$$ I know that $\sinh t$ can be represented as a series, but for that ...
3
votes
1answer
42 views

Power series of $\frac{1}{1+\frac{1}{4x}}$

Power series of $\frac{1}{1+\frac{1}{4x}}$ Now in an attempt to find this power series I used the known power series of: $\frac{1}{1+u} = 1-u+u^2-u^3+...$ Knowing this I simply substituted ...
0
votes
1answer
24 views

How do I work out the validity for a Maclaurin (power) series?

I cannot find the answer to this anywhere so I have decided to make a question. Given a Maclaurin series for a function, how can I quickly work out what the validity is for it? For example, ...
0
votes
1answer
29 views

Power series $\sum_{n=0}^\infty \frac{2n+3}{(2n)!}t^{2n}$

$\begin{align*} \sum_{n=0}^\infty \frac{2n+3}{(2n)!}t^{2n}&= \sum_{n=0}^\infty \frac{2n}{(2n)!}t^{2n}+ 3\sum_{n=0}^\infty \frac{t^{2n}}{(2n)!}=\left\{\begin{array}{c} 2n=k\\ n=0\Rightarrow k=0\\ ...
0
votes
0answers
10 views

Struggling with Frobenius Solutions

$x^2y''+5xy'+(x+4)y=0$ where $y = \sum_0^\infty c_n x^{n+r}$ a - prove $x=0$ is a regular singular point (done) b - find the r's (done) c - find the solution (stuck) also, I know the r's are both ...
1
vote
1answer
37 views

Finding series solution about zero

$y''+x^2y'+4y=1-x^2$ To find a power series, one substitutes in $y= \sum_0^\infty a_nx^n$. So after substitution, I've gotten $\sum_0^\infty (n+1)(n+2)a_{n+2}x^n + \sum_1^\infty (n-1)a_{n-1}x^n + ...
1
vote
2answers
48 views

Why does the taylor series of $\frac {1}{\ln x}$ have a non-infinite radius of convergence?

Shouldn't the taylor series of a function be equal to that function for any input value? Why does this not work for the taylor series of $\frac {1}{\ln x}$ when $|x| \gt 1$? Edit: I do mean the ...
2
votes
2answers
30 views

Interesting power series for $y'+y=\frac1x$

I had the differential equation $y'+y=\frac1x$, which I solved for $y$ as a power series: $$y=\frac1x\sum_{n=0}^{\infty}\frac{n!}{x^n}$$ Which was a power series at $\infty$, so it doesn't really ...
0
votes
2answers
35 views

When taking derivatives of power series, why do we shift the index up?

For example, if the series starts at n=0, and we take the derivative, the index usually then starts at n=1. This increases as we continue taking derivatives, but why do we need to do this? I get ...
0
votes
0answers
22 views

a function with many branch points : the radius of convergence of its Taylor series

How can I be convinced that if a (locally holomorphic) function $f(z)$ has many branch points, say at $\beta_1,\beta_2,\ldots,\beta_n,\ldots$, and all of the weirdest type, then the radius of ...
0
votes
0answers
12 views

Verification request- radius of convergence of two power series?

I need to find the region of convergence of : $\Sigma \frac{(n!)^2}{(2n)!}(x-2)^n$ $\Sigma \frac{x^{3n+1}}{(1+\frac{1}{n})^{n^2}}$ In 1- the series converges for $-2<x<6$, while in 2, the ...
1
vote
1answer
36 views

Change in Interval of convergence if center of convergence changes

So I have to find a power series that is centered at $-2^{1/2}$ If I choose to use the power series expansion for $e^x$ which converges for all $x$, and change $x$ to $x + 2^{1/2}$ does the interval ...
0
votes
1answer
25 views

Expanding $1/z$ about $z=-1$ using Taylor series vs Power Series

I need to expand $1/z$ about $z_0=-1$. I decided to do it using both methods, which don't agree. Using Taylor: Finding coefficients: $$f^{(n)}(z)=(-1)^n n!/z^{n+1} \Rightarrow f^{(n)}(-1)=-n!$$ ...
1
vote
1answer
34 views

Complex Taylor Series by substitution

I need to find the first few terms or so of the Taylor series centered at $z_0 = 0$ regarding these functions: a) $e^{z\sin z}$ b)$(1+z)^z = e^{z \ln (1+z)}$ c)$\cos (1 + z^3) $ d) $e^{e^z}$ ...
0
votes
1answer
45 views

Radius of convergence of $\sum\limits_{n}c_{n}z^{n^{2}}$ given that the radius of convergence of $\sum\limits_{n}c_{n}z^{n}$ is finite and nonzero

I know that the radius of convergence of a given power series $\sum_{n=1}^{\infty}c_{n}z^{n}$ is $R$, where $0<R<\infty$. Given this information, I need to find the radius of convergence of ...
0
votes
1answer
20 views

General form for complex limit function $\sum p(n) z^n$ where $p \in \mathbb{C} [n]$

Given a polynomial $p \in \mathbb{C} [n]$ of degree $k$, I need to show that the power series $\sum_{n=1}^{\infty} p(n) z^n$ uniformly converges in the open unit disc, and that the limit function $f$ ...
9
votes
4answers
392 views

Showing that $R(x)$ is a proper subset of $R((x))$ if $R$ is a field

I would like to show that if $R$ is a field, then $R(x)$ is a proper subset of $R((x))$, where $R(x)$ is the ring of rational functions, and $R((x))$ is the ring of formal Laurent series. If $f ...
1
vote
0answers
7 views

Transforming a polynomial sum using a series expansion (BCH codes)

In my study of BCH codes I've come across the following equation (the "key equation"): $$ \Omega(x) \equiv \Lambda(x)S(x) \mod x^{n-k} \tag{1} $$ Where the two terms on the right are defined by: $$ ...
0
votes
2answers
140 views

Complex power series : Radius of Convergence

Could anyone please suggest me how to deal with these questions (Complex Variable) : Note that all problems are in $\mathbb{C}$. 3.1 Determine the radius of convergence $\rho$ of each of the ...
1
vote
1answer
34 views

Radius of Convergence of Complex Power Series

I need to find the radius of the convergence of $\sum_{n=1}^{\infty}3^{n}z^{n^{2}}$ using the Cauchy-Hadamard formula. I'm not feeling 100% proficient at this method, however, so I'm asking 1) if what ...
0
votes
0answers
29 views

Show that $f(x)=\frac{1}{x^2}$ is real-analytic in $(0,∞)$

Show that:$$f(x)=\frac{1}{x^2}$$ is real-analytic in $(0,∞)$. I'm having trouble using Taylor's theorem to prove this
0
votes
1answer
19 views

function representation of power series

What is the function representation of this power series? [Summation from n=0 to infinity of ($x^n)(n+1)!/n!$ The solution is $\frac{1}{(1-x)^-2}$ but how??? I know that ...
1
vote
1answer
31 views

Find sum of power series. Having a small mistake.

Find the sum of the series. My answer is $-\frac{3}{4}$, but it should be $\frac{3}{4}$. Where did i make a mistake? $$ \sum_{n=1}^{\infty} \frac{n}{3^n} $$ $$ \frac{d}{dx} (\frac{1}{1-x}) = ...
0
votes
2answers
25 views

Abel's theorem power series

I am trying to show that if the power series $\sum (a_nx^n)$ coverges to a function f for $|x|<r$, then $$\int_{0}^{r} f(x)dx=\sum_{n=0}^{\infty} \frac{a_n}{n+1} r^{n+1}$$ provided that the series ...
0
votes
1answer
23 views

Finding the radius and the interval of convergence.

I usually use Ratio Test to find the radius and the interval of convergence. However, for this series, the ratio test does not work. If I use the ratio test, my answer is $|-2x+3|<1 $, ...
2
votes
1answer
52 views

Need help creating a power series for with specific condtions

I needed to give an example of a power series that satisfies the following conditions: interval of convergence is [$e$,$\pi$) I came up with this series: ...
0
votes
0answers
16 views

Confirmation on a function satisfying specific conditions(Power Series)

I had a question, find a function that satisfies the following conditions and I have to use Power series. F is the function. 1) Domain is all reals, 2) $F''(x) = cos(x^2)$, 3) $F'(0) = 3$, 4) $F(0) = ...
2
votes
1answer
17 views

Need help creating a power series with specific conditions

I needed to give an example of a power series that satisfies the following conditions: interval of convergence is [-1,1] and is conditionally convergence at both -1 and 1. Is it even possible to ...
8
votes
1answer
80 views

Trigonometric proof stuck with induction step

I am trying to prove: $$\sum_{s=0}^{\infty}\frac{1}{(sn)!}=\frac{1}{n}\sum_{r=0}^{n-1}\exp\left(\cos\left(\frac{2r\pi}{n}\right)\right)\cos\left(\sin\left(\frac{2r\pi}{n}\right)\right)$$ We know that ...
2
votes
1answer
156 views

Infinitely nested radicals

In a recent paper it was stated (and maybe proved) that we can solve any polynomial equation with nested radicals. Here "nested radicals" means expression such as: $$ ...
1
vote
2answers
38 views

Changing summation in a power series

I'm doing a question in my power series unit that involves adding summations together, I just started this unit so I'm not totally clear on how changing summation works, from what I understand you ...
0
votes
1answer
41 views

How to find an approximation of power series to correct to within $10^{-7}$ as faster?

If I approximate a $\displaystyle\int_{0}^{0.5} \frac{1}{1+x^7} dx$ correct to within $10^{-7}$. How to find it without using a calculator? Now I can't. I usually calculate every single term, and ...
2
votes
3answers
183 views

What is $2^{\frac{1}{4}}\cdot4^{\frac{1}{8}}\cdot8^{\frac{1}{16}}\cdot16^{\frac{1}{32}}\cdots\infty$ equal to?

I came across this question while doing my homework: $$\Large 2^{\frac{1}{4}}\cdot4^{\frac{1}{8}}\cdot8^{\frac{1}{16}}\cdot16^{\frac{1}{32}}\cdots\infty=?$$ $$\small\text{OR}$$ ...
1
vote
2answers
54 views

Sums of the series $1 + (x^2) / 3! +( x^4) / 5! +\cdots$

How can I compute sum of the series ; $$1 + \frac{x^2}{3!}+\frac{x^4}{5!}+\frac{x^6}{7!}+\frac{x^8}{9!}+\cdots$$ I tried to divide it to two pieces such that $$f(x) = ...
5
votes
2answers
36 views

Exponential Power Series where Powers are Prime

I am looking for information in regards to a couple particular functions: 1) $P(x)=\sum_{p\in\mathbb{P}}\frac{x^p}{p!}$ 2) $Q(x)=\sum_{p\not\in\mathbb{P}}\frac{x^p}{p!}$ (assuming $0, 1$ are ...
0
votes
0answers
22 views

Find a power series. centered at $x=-1$

I'm trying to find a power series, centered at $x=-1$ What do I have to do for next step? $f(x)=\frac{1}{2x-3} = \frac{1}{2(x+1)-2-3} = \frac{1}{5}[\frac{1}{\frac{2}{5}(x+1)-1}]$
0
votes
0answers
22 views

How to prove $\sum_{n\geq 0}{\frac{\Gamma(n+2+\alpha)}{n!\Gamma(2+\alpha)}z^n}=\frac{1}{(1-z)^{2+\alpha}}$

Let $z\in \{z\in\mathbb{C}:|z|<1\}, \alpha>-1,\Gamma(s)$ is the gamma function. How to prove $\sum_{n\geq 0}{\frac{\Gamma(n+2+\alpha)}{n!\Gamma(2+\alpha)}z^n}=\frac{1}{(1-z)^{2+\alpha}}$ ? If ...
0
votes
2answers
55 views

Prove that $|R(z)| \leq \frac{e-1}{(n+1)!}$ if $|z| \leq 1$ Complex Variables

Let $R(z)$ be the remainder after $n$ terms in the power series of $e^z$. That is $$R(z) = e^z - \sum_{k=1}^{n}\frac{z^k}{k!}=\sum_{k=n+1}^{\infty}\frac{z^k}{k!}$$ Prove that $|R(z)| \leq ...
1
vote
2answers
32 views

Prove the solution of $f''(x)-4f(x)=0$ is $f(x)=\sum_{p=0}^{\infty} \frac{4^{p+1}}{(2p)!}x^{2p}$

I'm wondering about this question : We have the differential equation $f''(x)-4f(x)=0$ and we want to find $f$ as a power serie with $f(0)=4$ and $f'(0)=0$. I would like to prove the only solution is ...
0
votes
0answers
15 views

For a power series, why is the value of n changed?

I tried to find the power series of the function. $f(x) = ln(10-x)$ During this, the value of n is changed. The first term is not zero. If n is changed, it is easy to calculate. Is there any other ...
0
votes
1answer
17 views

How toexpress $V=\frac{kq}{x-a}-\frac{kq}{x+a}$ in terms of $k,q,x,u$ in Taylor Series for the following condition?

The question calls $u=\frac{a}{x}$ and $u$ is the variable. So for Taylor Series, we express it in $f(x)=\sum^{\infty}_{k=0}\frac{f^k(0)}{k!}x^k$ However, one hint says all we need is geometric ...
1
vote
0answers
36 views

How to find function $F$ such that $F''(x)=\cos{x^2}$, $F'(0)=3$ and $F(0)=4$?

Here we want $F\in \Bbb{R}$. We use Taylor Series. I get $F''(x)=\cos{x^2}=\sum^\infty_{k=0}\frac{(-1)^k}{(2k+1)!}(x^2)^{2k}=\sum^\infty_{k=0}\frac{(-1)^k}{(2k+1)!}x^{4k}$ Integrating, we have ...
0
votes
0answers
91 views

How to find power series with the following interval of convergence?

$a) [-1,1] $ (conditionally convergent both at $-1$ and $1$) $b) [e,\ \pi) $ $c)$ center at $x=-\sqrt{2}$ and interval of convergence $(-\infty, \infty)$ I think to solve the question, we basically ...
1
vote
1answer
46 views

The uniform convergence of a power series on a connected open set

The problem is to prove that $$ \sum_{n=0}^\infty a_n(z-z_0)^n $$ uniformly converges on a connected open set $D$ if and only if it uniformly converges on $\overline{D}$. The sufficiency is ...
3
votes
0answers
54 views

“Logarithmic derivative of a p-adic number”

Coleman, in "Division Values in Local Fields", (Inventiones math. 53, 91 - 116 (1979)), says that In his work on cyclotomic fields Kummer observed that various formal operations on power series ...
1
vote
2answers
28 views

Use differentiation to find a power series

Use differentiation to find a power series of $f(x) = \frac{1}{(8+x)^2}$ $ f'(x) = \frac{-2}{(8+x)^3} $ how do I find the power series of this? I can not go next step.
0
votes
1answer
53 views

Find a power series.

Find a power series for the function. $f(x) = \frac {6+x}{1-x} = (6+x)*\sum_{n=0}^{\infty}x^n=6\sum_{n=0}^{\infty}x^n + \sum_{n=0}^{\infty}x^{n+1}$ What do I have to do for the next step? Thank you. ...
0
votes
1answer
39 views

Determine the first four non-zero terms in the power series expansion about $x=0$ for the general solution: $\left(2x-3\right)y''-xy'+y=0$

$$\left(2x-3\right)y''-xy'+y=0$$ First I found the first to derivatives of the following power series: $$y(x)=\sum_{n=0}^{\infty}a_nx^n$$ $$y'(x)=\sum_{n=1}^{\infty}na_nx^{n-1}$$ ...
0
votes
1answer
44 views

Does this complex power series converges on the unit circle?

Prove that the series $$ \sum_{n=1}^\infty \frac{(-1)^{[\sqrt n]}}{n}z^n$$ converges on $\partial B(0,1)$.Where $[x]$ implies the greatest integer that is not bigger than $x$. It is easy to prove ...
-1
votes
0answers
34 views

Solve $x^2f''(x)-x(2x^2-1)f'(x)-(2x²+1)f(x)=0$ using power series [duplicate]

I would like to solve $x^2f''(x)-x(2x^2-1)f'(x)-(2x^2+1)f(x)=0$ using power series, what is the way to go ? Thank you