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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4
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1answer
85 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
26 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
109 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 ...
0
votes
2answers
44 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
24 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
27 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
36 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
52 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 ...
1
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0answers
21 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
20 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
238 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
78 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
45 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
64 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
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2answers
38 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
88 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
34 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
31 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
30 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
35 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
49 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
45 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
47 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
81 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
49 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
707 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
32 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
68 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)} \\ &= ...
0
votes
1answer
23 views

Power series expansion requirements

Hello stackexchange folks :) I have a question regarding the assumptions made right before you choose to expand or approximate a function by a power series. Specifically I have the function: ...
1
vote
2answers
57 views

Justification for expanding exp(-x)/(1-exp(-x))

A geometric series $\sum{r^n}$ converges if $|r|<1$. In case $r = e^{-x}$, and needed $\int^b_0{\frac{x e^{-x}}{1-e^{-x}}}dx$ where $b>0$, how can I justify that is legal to make the series ...
1
vote
1answer
37 views

Power Series Representation of $x^3/(2-x)^3$

I don't need an answer, as this was a question I got wrong on a problem set, but could someone explain this? So, we have to represent f(x)= $x^3$/$(2-x)^3$ My professor writes consider g(x) = ...
2
votes
3answers
68 views

How do you add two series together

How do you add the series $$\frac{1}{2}\left(\sum_{n=0}^{\infty}\frac{2^{n}}{(z-3)^{n+1}} + \sum_{n=0}^{\infty}\frac{(z-3)^{n}}{4^{n+1}}\right)$$ ? is this right? $$\begin{aligned} ...
0
votes
1answer
17 views

Determing taylor series from other series

Consider $\cos(x)$ and $\cos(3x^2)$. How to determine the latter's Taylor series from the formers at $a = 0$? I'd write $$\cos{x} = \sum_0^\infty (-1)^n\frac{x^{2n}}{(2n)!}$$ Now, I could just ...
1
vote
1answer
60 views

formal power series expansion for square root

i want to prove this identity: $(1 + \sum\limits_{n=1}^\infty {1/2 \choose n} X^n)^2 = 1+X$ in the formal power series ring Q[[X]]. (so i can't just quote the binomial expansion for the square root) ...
3
votes
5answers
59 views

Formula for $r+2r^2+3r^3+…+nr^n$ [duplicate]

Is there a formula to get $r+2r^2+3r^3+\dots+nr^n$ provided that $|r|<1$? This seems like the geometric "sum" $r+r^2+\dots+r^n$ so I guess that we have to use some kind of trick to get it, but I ...
1
vote
1answer
29 views

Laurent series, function representation

Write the Laurent series for the function $f(z)=\frac{1}{1+z}$ $1<|z|<\infty$ I did $$\frac{1}{1-z}=\sum_{i=0}^\infty z^n\rightarrow \frac{1}{1+z}=\sum_{i=0}^\infty (-1)^nz^n$$ Is it right? ...
2
votes
1answer
36 views

Laurent series , function representation

Write the Laurent series around zero for the entire function $f(z)=z^2e^{3z}$ I'm a little confused on how to represent the complex functions by series, as I did in the calculation of real functions, ...
1
vote
4answers
59 views

Power Series of $\frac{3}{(1-3x)^2}$

The problem is to find the power series of this function $$\frac{3}{(1-3x)^2}$$ centered at $x = 0$. Normally you convert it into $\frac{1}{1-x}$ form. Since the denominator is squared do you ...
1
vote
1answer
45 views

Proving Convergence and Absolute Convergence of Power Series

How do you prove the following claim? If a power series $\sum_{n=0}^{\infty} a_n (x-a)^n$ converges at some point $b ≠ a$, then this power series converges absolutely at every point closer to $a$ ...
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1answer
30 views

The range of validity for the sums of power series

If I have a power series:($z$ is complex here) $\displaystyle \sum_{n=0}^{\infty}z^n$ valid for |$z|<1$ and another $\displaystyle \sum_{n=0}^{\infty}({\frac{z}{2}})^n$ valid for |$z|<2$ I ...
0
votes
1answer
25 views

Expressing $z\in\mathbb{C}[[w]]$ as a power series in $y\in\mathbb{C}[[z]]$.

I'm given that $$w=z+\sum_{i=2}^\infty a_iz^i$$ $$z=w+\sum_{i=2}^\infty b_iw^i$$ $$y=z-\sum_{i=2}^\infty (-1)^ia_iz^i$$ And that those series are all convergent (in particular I'm not given that they ...
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0answers
13 views

Convergence of integrated power series.

If I integrated a series that included the endpoints in the interval of convergence, will the new series also have those endpoints in its own interval of convergence?
1
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1answer
71 views

Compounded Interest with Exponentially Increasing Periodic Payments

Given the formula $$v_a = p\left(\frac{\left(1+\frac{r}{n}\right) ^{nt}-1}{\frac{r}{n}}\right)$$ for the value $v_a$ of an account growing at a periodic rate $r$ with a regular deposit $p$ compounded ...
1
vote
1answer
34 views

The autocovariance function of ARMA(1,1)

So I am reading Brockwell and Davis introduction to Time Series analysis on page 89 where he derives the ACVF of an $ARMA(1,1)$ given by: $X_t - \phi X_{t-1}=Z_t+\theta Z_{t-1}$ with ${Z_t}$ is ...
3
votes
1answer
50 views

$\sum_{n=0}^\infty z^n = \prod_{m=0}^\infty \left(1+z^{2^m}\right)$

When reading Iwaniec and Kowalski's Analytic Number Theory, I came across the following "identity" on page 11 (the Amazon link has a free book preview which includes page 11): $$\sum_{n=0}^\infty z^n ...
2
votes
2answers
47 views

How to evaluate the $\lim\limits_{x\to 0}\frac {2\sin(x)-\arctan(x)-x\cos(x^2)}{x^5}$, using power series?

How to evaluate the $\displaystyle\lim\limits_{x\to 0}\frac {2\sin(x)-\arctan(x)-x\cos(x^2)}{x^5}$, using power series? It made sense to first try and build the numerator using power series that are ...
3
votes
2answers
72 views

Inverse Rule for Formal Power Series

I am just really starting to get into formal power series and understanding them. I'm particularly interested in looking at the coefficients generated by the inverse of a formal power series: ...
0
votes
2answers
81 views

Example of a Power Series Given Interval of Convergence

This was a thought question assigned to our calc II class, and I wasn't sure how to approach it. Give an example of a power series whose interval of convergence is $(0, \frac{4}{3}]$. Show ...
1
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2answers
42 views

Power series properties. The sum of two series.

Let $\sum a_nz^n$ and $\sum b_nz^n$ power series with radiuses of convergences $R_1,R_2$ respectively. Suppose the radius of convergence of $(\sum a_n+b_n)z^n$ is $R$. Find an example in which ...
5
votes
1answer
104 views

Power series related to Bernoulli numbers

I'm reading Tenenbaum's Introduction to analytic number theory. He defines Bernoulli polynomials as the unique sequence $B_n$ such that $B_0=1$ $\forall n\geq0, B_{n+1}'(X)=(n+1)B_n(X)$ ...