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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3
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1answer
26 views

Field and ideal notation: double bracks/parens vs single brackets/parens

I'm reading some notes that has the following denotation: the set of formal power-series with coefficients in $\mathbb{F}_p$ is denoted by $\mathbb{F}_p[[t]]$. the fraction field, $\operatorname{...
0
votes
0answers
14 views

Sum of a converging series having Error function with a polynomial

I am struggling to find the sum of the following series: $k\sum\limits_{z=1}^{\infty} \frac{(z+1)^2}{4} . erfc(az)$ where $k$ and $a$ are known parameters and $erfc(x)$ is the complementary error ...
1
vote
1answer
16 views

Transition from convolution of PMF's to convolution of power series in a random walk

In the proof that symmetric random walks end up regressing to the origin with probability $1$, I have found this didactic post on-line. In it the following two definitions are given: Probability of ...
0
votes
1answer
25 views

Finding Taylor series without using derivatives

If $\displaystyle f(z) = \frac{e^{iz}}{z^2-1}$ then we can set $g(z)=e^{iz}$ and $h(z)=z^2-1$. The Maclaurin expansion for $e^{iz}$ is $$\sum\limits_{n=0}^\infty \frac{(iz)^n}{n!}$$ so $\displaystyle ...
0
votes
1answer
25 views

Random walk - probability of first pass through zero.

In the proof that symmetric random walks end up regressing to the origin with probability $1$, I have found this didactic post on-line, where the power series of the probability mass function of the ...
1
vote
2answers
48 views

How to use the generalized binomial theorem to produce the power series of $(1-x)^{1/2}$ [duplicate]

I am trying to see how to get from $\sqrt{1-x}$ to the power series $\displaystyle\sum_{m=0}^\infty\frac{-1}{2m-1}\,{2m \choose m}\,\frac{x^m}{4^m}$, ideally using the generalized binomial theorem. I ...
0
votes
0answers
15 views

Asymptotic Expansions of a Generalized Hyper-Geometric Function

Let $t>0,x>0$, and $$\{a_1,a_2,a_3\}=\{2, 2, 9/8 - (i t)/2\}$$ $$\{b_1,b_2,b_3,b_4\}=\{1, 1, 3/2, 17/8 - (i t)/2\}$$ We are looking for the asymptotic expansions of a generalized hyper-...
0
votes
0answers
39 views

Deciphering the theorem of perfect powers

Can someone help me understand what happened to this equation from the paper entitled Perfect Powers With All Equal digits but one...I don't understand the part when it lets a and c not equal to zero....
0
votes
0answers
28 views

Using a geometric series to calculate the sum of a power series-what is wrong?

Assume I need to calculate the sum of the following power series: $$ S=\sum_{I=1}^\infty \frac{(-1)^n x^n y^n}{n} $$ The first way is to substitute: $$ t=xy \Rightarrow S= \sum_{I=1}^\infty \frac{(-1)...
7
votes
2answers
112 views

Stationary distribution of “probabilistic geometric series” with two alternative ratios

I have an iterative process starting at $X_0=2$. In each iteration $i=1,2,\ldots$, the value of $X_i$ is determined based on the value of $X_{i-1}$ as follows: With probability 0.5, $X_i=qX_{i-1}+1$, ...
0
votes
1answer
30 views

Interval of convergence of $\sum_{k=0}^\infty$ $(-1)^k$ $(2x)^{2k}$

Given $\sum_{k=0}^\infty$ $(-1)^k$ $(2x)^{2k}$, I have to evaluate the interval of convergence. Approach I have to evaluate the convergence radius. Using Cauchy-Hadamard, I receive: $\...
3
votes
1answer
34 views

Power series representation of $f(x) :=$ $1 \over {1 + 4x^2}$

Let $f:$ ($-1 \over 2$, $1 \over 2$) $\rightarrow \Bbb R$, $f(x) :=$ $1 \over {1 + 4x^2}$ be a function. I am searching for a power series representation of this function. I ...
0
votes
1answer
33 views

Evaluate convergence radius for $\sum_{n=0}^{\infty}$ $(3x - 2)^n \over 5^n(n+2)\sqrt{n+3}$

Follow-up question (see "Edit") Given $f(x) :=$ $\sum_{n=0}^{\infty}$ $(3x - 2)^n \over 5^n(n+2)\sqrt{n+3}$, I have to evaluate the largest open interval where $f(x)$ converges. ...
0
votes
1answer
38 views

Why does this work to shift a power series?

Problem: Find the Taylor series and the interval on which it is valid for $f(x) = \frac{1}{1-x}$ centered around $x=5$. The textbook's solution says to write $$\frac{1}{1-x} = \frac{1}{-4-(x-5)} = -\...
0
votes
1answer
32 views

How to prove that $h''(x)$ has at most one zero on $(0,1)$.

$h(x)=1-\sum_{i=1}^{k-1}x^i+a_kx^k+\sum_{i=k+1}^\infty x^i$, where $|a_k|\le1$, is the power series of an analytic function. Prove that $h''(x)$ has at most one zero on $(0,1)$.
2
votes
1answer
45 views

Convergence of a cosine series

Given that $$\sum_{n=1}^\infty a_n<\infty,$$ and that $$\lim_{n\to \infty}b_n=0$$ Is the series $$\sum_{n=0}^\infty a_nb_n^{-2}(1-cos(b_n))$$ necessarily convergent?
1
vote
1answer
14 views

Singularities of quotient of polynomials where the degree of the denominator $\ge$ the degree of the numerator $+2$.

Let the degrees of the polynomials $$P(z)=a_0+a_1 z+a_2 z^2+\cdots +a_n z^n \; (a_n \neq 0)$$ and $$Q(z)=b_0+b_1 z+b_2 z^2+\cdots +b_m z^m \; (b_m\neq 0)$$ be such that $m \ge n+2.$ Show that if ...
2
votes
1answer
46 views

Showing that $c_{i}\equiv 0\pmod{p}$

Let the numbers $c_{i}$ be defined by the power series identity $$\frac{1+x+x^{2}+\ldots+x^{p-1}}{(1-x)^{p-1}}= 1+c_{1}x+c_{2}x^{2}+\ldots$$ Show that $c_{i}\equiv 0\pmod{p}$ for all $i\geq 1$. $\...
0
votes
1answer
26 views

Algebraic series, rational fraction of two variables in the form of polynomial

I come across the following claim: Let $y\in\mathbb{C}[[x]]$ be an algebraic series, that is, there exist $n\in\mathbb{N}^*$ $A_i(x)\in\mathbb{C}[x]$ for $i=0,...,n$ and $A_n(x)\neq 0$ such that \...
1
vote
3answers
75 views

Find the power series representation of $e^{-x^2}$

I know that the Maclaurin expansion of $e^x$ is $$1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...$$ But i'm not sure how to find the Maclaurin series here I tried this $$ f'_{(0)}=-2xe^{-x^2}=0 $$ And that ...
2
votes
1answer
47 views

Is it possible to express the inverse of a polynomial as a series?

Is it possible to express the multiplicative inverse of a polynomial in descending powers of n i.e. \begin{equation} \frac{1}{\left[\sum_{k=0}^\infty a_kt^{n-2k}\right]^2} \end{equation} as a series ...
2
votes
3answers
64 views

Why does changing the center of a geometric power series change the interval of convergence?

I know that the interval of convergence of the geometric power series $$\sum_{n=0}^\infty x^n=\frac{1}{1-x}$$ is $(-1,1)$. Why is it that if I do the following manipulation $$\frac{1}{1-x}=\frac{1}{...
2
votes
2answers
54 views

Representing $\ln(x)$ as a power series centered at $2$ without computing any derivatives

I am working through a calc book and one of the problems asks the above question. However, taylor and maclaurin series have not been introduced yet. In some worked examples, they leverage old series,...
0
votes
5answers
148 views

What is the sum of the series 1/3 + 2/9 + 3/27 + 4/81 + … [duplicate]

I remember solving this in highschool , but now I don't remember how to find sum of these kind of series . I want to find the sum of the general series Sum $\sum_{n=1}^{\infty} n .a^{-n} = ? $ ...
0
votes
2answers
49 views

proving theorem about perfect powers

Im currently studying the journal entitled Perfect Powers with All Equal Digits but One theorem: For a fixed integer $l \geq 3$, there are only finitely many perfect $l$-th powers all whose digits ...
1
vote
0answers
15 views

Radius of convergence for the following power series using the ratio test

I am slightly unsure about how to do the following question relating to the radius convergence (using specifically the ratio test). The power series is as follows: $$\sum_{n=1}^{\infty}\frac{(2x+1)^n}...
-5
votes
3answers
141 views

Which function of $x$, other than $x +c$, and Integral of ($\cos x)^2+(\sin x)^2$, and Integral of $e^{iPi}$ has derivative = 1. [closed]

It is a simple question: Which function of x, other than x +c, and Integral of (cosx)^2+/(sinx)^2, and the integral of $-e^{i*Pi}$, has derivative =1.
0
votes
1answer
46 views

Writing one formal power series as a function of another

Suppose we have a formal power series $x(t)=t+\sum_{k=2}^\infty x_k(t^k/k!)$. In principle, this can be inverted to obtain $g(x)=x+\sum_{k=2}^\infty g_k(x^k/k!)$ such that $x(g(x))=x$. The specific ...
2
votes
0answers
67 views

Series expansion of inverse polynomial

Suppose an nth order polynomial $P_n(x)$ with real and distinct roots $d_1,d_2,\dots,d_n$, which has the factorization \begin{equation} P_n(x)=(x^2-d_1^2)(x^2-d_2^2)\cdots(x^2-d_n^2).\end{equation} ...
0
votes
1answer
43 views

Sum of a series with exponential and polynomial terms [closed]

I have reduced the expression that I am working on to the following sum of series, which is definitely converging. It would be great if someone can help me out with this or suggest ways this can be ...
0
votes
2answers
51 views

Power series as approximation

I have to estimate the error when I approximate the function $$e^{\sin x}$$ to $$1+x+x^{2}+x^{3}$$ when $|x|<0.1$. I really don't know how to do because my teacher didn't teach me. But what I did ...
1
vote
2answers
42 views

Find the Maclaurin series of f(x)=(arctan(x)-x)/x^3

What I think I need to to do is find a general series expansion of the function and then derive term by term to get the Macaurin series...but I'm not quite sure how to expand this function. Any help ...
2
votes
1answer
42 views

All power series has a point that is not regular.

Definition: Let $f = \sum_{n \geq 0} a_n z^n $ a power series and $0<R< \infty$ its convergence ratio. We say that $z_0 \in \mathbb C, |z_0| = R$ is a regular point if $\exists r > 0$ such ...
3
votes
1answer
209 views

help verifying equation $\int_0^ x \frac{1}{1+t^n} dt$

As a follow up to a previous posting addressing the integral of $1/ (t^n+1)$ for $n\in \Bbb{N}$ I found the following $$\int_0^ x \frac{1}{1+t^n}\, dt=\sum_{i=0}^{\infty}\frac{(i!)(n^i)x^{in+1}} {(x^...
0
votes
1answer
71 views

Simple Power Series Solutions for advanced applied math

First of all, the image is my question page. Then, I have my answer for this question just don't know if it is right or not. If I am wrong, can you give me a suggestion?
1
vote
1answer
43 views

Interval of convergence for series with complex numbers

I'm trying to find interval of convergence of this series: $$\sum_{n=1}^{\infty} \frac{7^n(z+2i)^n}{4^n+3^ni}$$ and I should draw a plot which represents the answer, this is what I've got so far: ...
0
votes
1answer
64 views

Power series / Taylor series approximation

I need to find $k\in N$ such that $$ \dfrac{k}{10\ ^ 5} \le \arctan(0.1) \lt \dfrac{k +1}{10\ ^ 5} $$ I tried using Lagrange Remainder formula to find that k but with no luck. If I am using ...
0
votes
1answer
37 views

Radius of Convergence for Polynomial

I have to find the radius of convergence for this one, but I haven’t found a solution for this type of a term. $\mathbb f (x)$ = $1 + 7x^3 + 5x^4 + x^{13} + x^{2015}$ Could you help me out?
0
votes
1answer
38 views

How to solve: $\ \lim_{n \to +\infty} \frac{n^n + \frac {1}{n}}{(n + \frac {1}{n})^n} \ t^n $

How can I solve: $$\ \lim_{n \to +\infty} \frac{n^n + \frac {1}{n}}{(n + \frac {1}{n})^n} \ t^n $$ tis a whole number. Thank you very much! Please tell me your ...
-5
votes
1answer
124 views

Telescoping function Revealed.

Part B: I found Summation $$\sum_{n=0}^\infty\frac{x^n \ (-1)^n}{(y+1)^{n+1}} = \frac{1}{(x+y+1)}$$ However $x$ is related to $y$. $y \ge |x|$. You may derive the result and present it by ...
4
votes
1answer
207 views

New series formula for $\arctan(x)$?

I discovered this equation, but have no idea if it has been previously discovered. Please help determine if it has been previously developed. Or please prove that the equation is not correct. $$\...
1
vote
0answers
34 views

Laurent-Series on an annulus

I solved an exercise and I want to know if it is correct. I'm trying to find the Laurent-Series for $$g(w)=\frac{w}{1+w^2}$$ On the annulus $D_{1,2}(-i)$ What I did so far: We have 2 poles in $w=i$ ...
1
vote
0answers
37 views

Singular expansion of an implicit function

In the book of Flajolet and Sedgewick (this context is not so important, though), the following argumentation is used: Let $y(z)$ be a function given implicitly by $y - \phi(z,y) = 0$, where $\phi$ ...
0
votes
2answers
27 views

$\sum _{j=0}^{\infty }\binom{-p-1}{j} \bigl( -\frac {x} {1+x}\bigr) ^{j}=?$

I did try to use geometric series somehow. I have no idea how to evaluate in terms of $p$ and $x$.
2
votes
1answer
21 views

Computing the taylor series of $f(x)=(x+2)^{1/2}$ around $x=2$ up to order 2 terms using binomial series

I can expand $(x+2)^{1/2}$ by taking $y=x+1$ and using the binomial series to find a power series representation of $f$: $$\sqrt{y+1}=\sum_{k=0}^{\infty}{1/2\choose k}y^k=1+\frac{y}{2}+\frac{\frac{1}{...
0
votes
1answer
29 views

Asymptotically equivalent series for uniform convergence

I have to find sets of uniform convergence of $$\sum_{n=1}^{\infty}n^2 \sin \frac{x}{n^4}$$ what if I study this series passing to the asymptotically equivalent $$\sum_{n=1}^{\infty}n^2 \frac{x}{n^4}$$...
5
votes
2answers
59 views

Deriving the additive property of natural log from power series?

I know the additive property of logarithms, that $$\ln(x) + \ln(y) = \ln(xy)$$ is easy to prove using the logarithm's nature as the inverse of the exponential function. However, I'm interested in ...
1
vote
1answer
68 views

Summing power series $\sum_{n=1}^\infty \frac{(2n-1)!!}{(2n)!!}x^n$

Lets have series $$\sum_{n=1}^\infty \frac{(2n-1)!!}{(2n)!!}x^n$$ Obviously, its convergence radius is 1. I should sum it, but don't know what's up with the double factorial. There is a hint in the ...
3
votes
2answers
137 views

Prove this series does not uniformly converge

$$\sum_{n=1}^{\infty}n^2 \sin \frac{x}{n^4}$$ It is easy to show that it absolutely converges. But what about uniform convergence? With M-test: $$|| f_n|| = \sup (| n^2 \sin \frac{x}{n^4}|) \leq \...
-1
votes
3answers
67 views

Is the following is true? If that so, give me a proof. $-log(1-x)=log(1+e^x)$??

Is the following is true? If that so, give me a proof. $$-log(1-x)=log(1+e^x)?$$ Give me some value where this equality holds. I dont think so it will be same. Because, $$(1-x)^{-1}=1+x+x^2+x^3+\...