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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1answer
41 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.
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0answers
28 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
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0answers
32 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
80 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
78 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
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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
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3answers
74 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 ...
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2answers
86 views

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

Simplify $(\cos x)^{2^{x^{\cos x}}}$ with respect to $x$ & $pi$... if $x > 0$ and $cos(x)$ $> 0$
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2answers
35 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
21 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
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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
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2answers
46 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 ...
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0answers
32 views

Use power series to approximate the definite integral $\int_0^{0.5} \frac{1}{1+u^5}\,du$ [closed]

Use power series to approximate the definite integral with an error less than $0.000005$: $$\int_0^{0.5} \frac{1}{1+u^5}\,du$$ Can you please walk me through it/ explain the concept? I'm having a ...
0
votes
1answer
87 views

Combinatorial Power Series proof [on hold]

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
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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
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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
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3answers
218 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
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1answer
42 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
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0answers
41 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
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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?
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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
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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
19 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
33 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
34 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
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3answers
30 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
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1answer
35 views

Complex Series proof

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}, ...
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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
44 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
619 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
65 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
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1answer
20 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: ...
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1answer
46 views

what are these Analytic functions? [closed]

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are ...
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2answers
51 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
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1answer
35 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
65 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
34 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
54 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
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1answer
18 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
29 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
43 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 $a$ ...
1
vote
1answer
24 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
24 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 ...
0
votes
2answers
33 views

Impossible interval for convergence of a power series. [closed]

For the series $$\sum_{n=0}^{\infty} a_n(x-2)^n$$ Which of the following intervals is impossible for the convergence of that series? $$1≤x≤3$$ $$1≤x≤4$$ $$0≤x≤4$$ $$-1≤x≤5$$