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
25 views

How do we derive the sum of $3^n$ and $2^n$

I know that $\quad\sum2^n = 2 (2^n-1)$ How can we derive this summation? And also how can we deduce the summation of $3^n$ from this ? I did observe this pattern : $$ \begin{align} n &= 1 ;\ ...
2
votes
0answers
6 views

Series $\sum_{d \geq 0} \sqrt{d} \cdot z^d$

As stated in the title, I'd like to compute the series $\sum_{d \geq 0} \sqrt{d} \cdot z^d$ where $z$ is a (small enough) complex number. More generally, for any real value $\alpha$, is there a ...
0
votes
1answer
28 views

The power series $\sum_{n\geq 1} \frac{x^{n}}{n(2n-1)}$ with $2$nd Taylor polynomial and Taylor series.

Consider the series $$\sum_{n\geq 1} \frac{x^{n}}{ n(2n-1)}\tag{1}$$ and the sum is denoted by $f(x)$. i) I want to determine the second Taylor polynomial for $f(x)$ about $x=0$. ii) I want to ...
1
vote
0answers
33 views

Weird question about interval of convergence

The question is: if $$ f(x) = \sum \limits_{n=0}^\infty x^n$$ determine the interval of convergence for the power series representation of $$\int_0^x f(t) \, dt$$ That integral threw me off.
1
vote
1answer
28 views

Power series representation?

The function to represent as a power series is: $$ \frac {10} {(x-10)^2} $$ Any help is, as always, appreciated.
0
votes
1answer
25 views

Interval of convergence of power series?

If the power series is: $$ \sum\limits_{n=1}^{\infty}\frac{x^n}{\sqrt{n+1}} $$ and I've found the interval to be $$ -1 < x < 1 $$ then would the answer $$ (-1, 1) $$ work? some other options ...
2
votes
3answers
29 views

(Simple question) Radius of covergence of power series?

For the power series: $$ \sum\limits_{n=1}^{\infty}\frac{(x-1)^n}{2^n} $$ Would radius of convergence be $$ x = 1 $$ ?
0
votes
1answer
29 views

Determine the radius of convergence of the power series

Determine the radius of convergence of the power series $\sum \limits _{n=4} ^\infty \frac {2n+4} {4^{n+5}} (x-8)^{4n+1}$. I tried the ratio test to find where $\frac {a_n} {a_{n+1}} < 1$ but I ...
2
votes
1answer
41 views

What is the center of power series?

The power series is: $$ \sum\limits_{n=1}^{\infty}\frac{(x+4)^n}{n+1} $$ Any help appreciated!
2
votes
0answers
35 views

Does convergence of power series on radius of convergence imply absolute convergence?

Let $R$ be radius of convergence of power seires $\displaystyle\sum_{k}a_kz^k$. If the power series converges for all $|z|=R$, can we say that it converges absolutely on the radius of convergence? I ...
0
votes
1answer
23 views

Show that if $f(z)=\frac{\operatorname{Log}z}{z-1}$ when $z\neq 1$ and $f(1)=1$, then $f$ is analytic throughout the domain.

$\operatorname{Log}z=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}(z-1)^n \; (|z-1|\lt 1).$ Use this fact to show that if $$f(z)=\frac{\operatorname{Log}z}{z-1} \; \text{when} z\neq 1$$ and $f(1)=1$, ...
2
votes
1answer
38 views

Convergence of complex power series $z^{n!}$ at boundary

I'm revising for an exam at the moment and I'm struggling with part of a question. I'm asked to find the radius of convergence of the series $\sum_{n=0}^{\infty }z^{n!}$ and then find where it ...
0
votes
1answer
33 views

With the aid of series, show that if $f(z)=\frac{\operatorname{cos}z}{z^2-(\pi/2)^2}$, then $f$ is an entire function.

Prove that if $$f(z)= \begin{cases} \frac{\operatorname{cos}z}{z^2-(\pi/2)^2}, & \text{when} \; z\neq \pm \pi/2, \\ -\frac{1}{\pi}, & \text{when} \; z=\pm \pi/2, \end{cases} $$ then $f$ is ...
3
votes
1answer
87 views

Is $\mathbb{Q}(\pi) \cong \mathbb{Q}[[x]]/ \langle \sin(x) \rangle$?

If we let $\mathbb{Q}[[x]]$ be the set of all power series with rational coefficients then can we say that $\mathbb{Q}(\pi) \cong \mathbb{Q}[[x]]/ \langle \sin(x) \rangle$?
0
votes
1answer
19 views

“Sum of power” for prime numbers

I use Euler–Maclaurin formula, Faulhaber's formula and Bernoulli polynomials for "sum of powers" for this type $\sum_{t=1}^nt^k$. but I don't know to find compact form when sum is taken from first ...
2
votes
1answer
43 views

Show that the binomial series satisfies: $(1+x)f'(x)=\alpha f(x)$

If $f(x) = \sum_{n=0}^{\infty}{\alpha\choose n}x^n$, show (without assuming the results of the binomial theorem) that $$(1+x)f'(x)=\alpha f(x)$$ for $|x|<1$ I've already shown that the sum ...
0
votes
1answer
45 views

Power Series Expansion

How can I find the Maclaurin series for $f(x)=e^x$/$(1-x^2)$? I have tried expanding it out but I am having trouble with the algebra of it.
0
votes
0answers
37 views

Computing an exponential generating function from the first few terms

The current question is related to this one, and this other one. I have a number sequence, and I want to find generating ...
4
votes
0answers
34 views

Questionable Convergence of a Series

The summation is: $$ S = \sum_{k \geq 0} f(k) \int_{0}^{\pi/2} \sqrt{1-(1- \frac{f(k+1)^2}{f(k)^2})\sin^2(\theta)}d\theta $$ Now, we know that $f(k+1) < f(k)$ and as $k$ approaches infinity, ...
3
votes
1answer
85 views

Power series as fractions

This is what I did: \begin{equation*} (x^3-x^6)x^6[x+x^2+x^3+..], \\ \frac{(x^3-x^6)x^6}{1-x}. \end{equation*} What mistake did I make? And, How to solve this: $1+3x^2+9x^4+27x^6+...+3^{157}x^{314}$ ...
4
votes
1answer
79 views

How many $s,t,u$ satisfy: $s +2t+3u +\ldots = n$?

Given $n\in \mathbb{N}^+$, what is the possible number of combinations $s,t,u,\ldots\in\mathbb{N}$, such that: $$s +2t+3u +\ldots = n\quad?$$ Additionally, is there an efficient way to find ...
1
vote
0answers
24 views

Decomposing a series

When I insert the following function \begin{equation} F(X,Y)=-\frac{1}{Y^{2/3}}\sum _{m=0}^{\infty } \frac{\Gamma \left(\frac{m+2}{3}\right)}{m! \Gamma (m+1)}\left(-\frac{X^2}{2^2 ...
3
votes
3answers
99 views

How to show that $\sum_{k=0}^{\infty} \frac{x^{k}}{k!}$ represents a continuous function

(This is a homework problem) I am trying to show that the series $\sum_{k=0}^{\infty} \frac{x^{k}}{k!}$ represents a continuous function on $\mathbb{R}$. My idea was to show that the functions ...
-2
votes
2answers
86 views

Simplify $(\cos x)^{2^{x^{\cos x}}}$ with respect to $x$ & $pi$ [closed]

Simplify $(\cos x)^{2^{x^{\cos x}}}$ with respect to $x$ & $pi$... if $x > 0$ and $cos(x)$ $> 0$
0
votes
2answers
39 views

Finding the radius of convergence of a power series, $\sum_{n=1}^{\infty} a_n x^n$.

I have to detemernine the radius of convergence of the power series $\sum_{n=1}^{\infty} a_n x^n$, where $(a_n)_{n=0,1,2,...}$ is given by $a_n=2-\dfrac{1}{2}a_{n-1}$ with $a_0=2/3$. So far I've ...
0
votes
1answer
22 views

ODE Series Solution

For the ODE: $$\frac{dy}{dx}=2y$$ If the successive derivatives calculated are: $$y'=2y,y''=2y'=(2^2)y,y^{(3)}=(2^3)y,\ldots,y^{(n)}=(2^n)y$$ How do I find the coefficients of the following ...
1
vote
1answer
22 views

First order approximation of $F(x)=\int_0^x f(t) dt$ in the neighbourhood of $\infty$

Let $f(x)$ continuous on the real line. Then the first order approximation of $$F(x)=\int_0^x f(t) dt$$ in the neighbourhood of $0$ is: $$F(x)=\int_0^x f(t) dt\sim 0 + x f(0), \ \ \ (x\rightarrow 0)$$ ...
1
vote
2answers
32 views

Periodicity of trigonometric functions directly from their power series

My question is very simple yet I've gotten nowhere with it. Is there any way one can, without directly or indirectly referencing any differential equations satisfied by the circular trigonometric ...
1
vote
2answers
47 views

Radius of convergence of $\sum_{n\geq 0}a_{n}x^{n}$.

Consider a series $\sum_{n\geq 0}a_{n}x^{n}$ where $a_{0}=2/3$ and $a_{n}=2-(1/2)a_{n-1}$ for all $n$. It is assumed that $2/3\leq a_{n}\leq 5/3$ for all $n\geq 1$. My problem is about determining its ...
0
votes
1answer
88 views

Combinatorial Power Series proof [closed]

Need help proving the following involving power series $A(x)$ and $B(x)$: If $A(x)B(x)=0$ (the power series where every coefficient is 0), then $A(x)=0$ or $B(x)=0$. AND If $(A(x))^2=(B(x))^2$, ...
1
vote
0answers
20 views

Is this statement about Abelian/Tauberian theorems true?

Suppose we have some real constants $c_n \geq 0$, and know that $$\sum_{n=0}^{\infty} c_nr^n$$ converges for all $r \in (0,1)$. Suppose that the limit $$\lim_{r \uparrow 1} (1-r)\sum_{n=0}^{\infty} ...
1
vote
1answer
19 views

How do I find the set $U$ on which this series defines a holomorphic function?

I have just come across a question that asks me to find the set $U$ on which this series defines a holomorphic function. I have trawled through my notes but I can't find anything, any help on how I ...
0
votes
3answers
222 views

Series Solution of an ODE

The ODE below is required to help compute the coefficients of function. There isnt any information about this topic in my textbook so i am just wondering how i would go about this question? In this ...
1
vote
1answer
59 views

limit of jacobi theta 2 or simple series

I have a simple problem: I need to evaluate the limit $x\rightarrow 1$ of the Jacobi Theta function 2 $$\Theta_2(m,x)=2x^{1/4}\sum_{k=0}^\infty x^{k(k+1)}\cos((2k+1)m)$$ when $m=0$, that to say ...
0
votes
0answers
42 views

Fourier Expressions

In the Fourier series, what are all the ways we can express: $\displaystyle\sin\left(\frac{n\cdot\pi}2\right)$ $\displaystyle\cos(n\cdot\pi)$ I know we can express as $(-1)^{(n+1)}$, and as ...
0
votes
1answer
62 views

What are the four last numbers in the series $1^1 + 2^2 + 3^3 +\cdots+3458^{3458}$?

What are the four last numbers in $1^1 + 2^2 + 3^3 +\cdots+3458^{3458}$ Hello, I have come across this question, and I have no idea how to solve it. What do you guys think?
1
vote
2answers
36 views

Evaulate/approximate a series formula $\sum_{i=1}^{n}\left ( \frac{1}{n}\right)^i \left(\frac{n-1}{n}\right)^{n-i}$

Given a fixed $n$, we define two probabilities $p_1=\displaystyle \frac{1}{n}$ and $p_2=1-p_1 = \displaystyle \frac{n-1}{n}$. The goal is to evaluate/approximate $\displaystyle \sum_{i=1}^{n} p_1^i ...
1
vote
1answer
69 views

$f(x)=\sum_{i=0}^{\infty} (x^{2^n})/(1-x^{2^{n+1}})$. Find $f(99)$.

$f(x)=\sum_{i=0}^{\infty} (x^{2^n})/(1-x^{2^{n+1}})$. Find $f(99)$. ATTEMPT: The following series can be re-written as $f(x)=\sum_{i=0}^\infty \left(\frac{1}{1-x^{2^n}}\right) \cdot \left( ...
0
votes
1answer
21 views

Series expansions of inverse polynomials

Suppose one is given a strictly monotonous polynomial, $$f(x) = \sum_{n=0}^N a_n x^n$$ So that for a given $y$ there exists a single real $x=f^{-1}(y)$. It would be nice* to be able to calculate the ...
1
vote
1answer
23 views

Applying the Frobenius method to $x^2 y'' - 2x y' - 10y = 0$

Here is the equation: $$x^2 y'' - 2x y' - 10y = 0 \tag{E}$$ We want to find, using the method of Frobenius, a solution in the neighbourhood of $0$, which is here a regular-singular point. ...
4
votes
2answers
28 views

Question about radius of convergence.

I want to determine the radius of convergence of the series \begin{equation*} \sum_0^\infty \frac{f^{k}(5)}{k!}(z-5)^k, \end{equation*} where $f(z) = \frac{z^2}{e^{iz}-1}$. In the solution of ...
0
votes
1answer
34 views

Expand $(e^{2x}-1-2x)/x^5$ into Laurent Series on 0<|x|<$\infty$ and classify its singularity

I guess I'm having difficulty with this because its not in the form of a polynomial expression, which is what I've been taught. Nevertheless here's what I did: I know that the expansion for ...
0
votes
1answer
39 views

Complex number, power series

Develop $\sinh z$ in powers of $z-\pi i$ to show that $$\lim_{z\to \pi i}\frac{\sinh z}{z-\pi i}=-1$$ I know that $\sinh z=\sum_{n=1}^\infty \frac{z^{2n-1}}{(2n-1)!}$. Edit: Following the hint ...
1
vote
3answers
34 views

Complex number, entire function

Let $f(z)=\frac{(e^{cz}-1)}{z}$ if $z\neq0$ and $f(0)=c$ show that f is entire Theorem:A power series represents a analytical function inside their circle of convergence. I know I could prove ...
0
votes
1answer
45 views

Integrating $\frac{1}{1+z}$ along a path to derive the Maclaurin series for $\mathrm{Log}(1+z)$

Integrate the Maclaurin series for$\frac{1}{1+z}$ along a path, inside the circle of convergence, going from $z'=0$ to $z'=z$ and show that $$Log(z+1)=\sum_{i=1}^\infty (-1)^{n+1}\frac{z^n}{n}, ...
-2
votes
1answer
44 views

Finite power series [duplicate]

I'm a student and I'm looking for a solution for the following finite power series: $$ \sum_{n=0}^m \frac{1}{n!} x^n $$ By "solution" I meant expansion of the series and finding a closed form ...
3
votes
1answer
45 views

Interesting Power Series

The series is $\sum_{n=1}^{\infty} r(n)x^n$ , where $r(n)$ is defined as the divisor function. The question is , what is the radius of convergence of the power series? Maybe it is not that interesting ...
5
votes
4answers
626 views

Number raised to power of irrational number

What is the consequence of raising a number to the power of irrational number? Ex: $2^\pi , 5^\sqrt2$ Does this mathematically makes sense? (Are there any problems in physics world where we ...
0
votes
1answer
31 views

Convergence radius of complex power series

If $a_n\neq 0$ for all $n \geq n_0$ and $\lim|\frac{b_n}{a_n}|=1$, then $\rho(S)=\rho(T)$. Since S=$\sum a_nz^n$ and T=$\sum b_nz^n$. I tried to use the definition of convergence radius $$\limsup ...
4
votes
1answer
66 views

What did i do wrong with this derivation?

$$ \cos(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} $$ Therefore \begin{align} \frac{1}{\cos(x)} &= \frac{1}{1-(\frac{x^2}{2} - \frac{x^4}{4!} + \frac{x^6}{6!} - \cdots)} \\ &= ...