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
10 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
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1answer
44 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 $i\geq 1$. $\...
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1answer
20 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 \...
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3answers
71 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
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1answer
29 views

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

Is it possible to express the multiplicative inverse of a nth order polynomial i.e. \begin{equation} \frac{1}{\left[\sum_{i=0}^na_ix^i\right]^2} \end{equation} as a series using binomial theorem or ...
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2answers
38 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}{...
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2answers
51 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,...
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0answers
29 views

Convolution form and $n$-th partial term of $\sum_{k=1}^{+\infty} kp^k$ [on hold]

With $p\in]0,1[$, let $\displaystyle x(k)=\sum_{m=1}^{+\infty}m$ and $\displaystyle \phi(k)=\sum_{m=1}^{+\infty}p^m\:$; $m,k\in\mathbb{N}$ So that by the convolution product, $\displaystyle (x*\phi)(...
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5answers
138 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} = ? $ ...
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0answers
21 views

Approximation of elliptic integral

I am trying to find an approximation (series expansion) of the elliptic integral \begin{equation}I_n(x)=\int_x^{+\infty}\frac{dt}{(E_n(t))^2\sqrt{t^2-a^2}\sqrt{t^2-b^2}}\end{equation} where $E_n(t)$ ...
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2answers
30 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 ...
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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}...
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3answers
131 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. [on hold]

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. Think Kepler's third law is a constant. [an ...
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1answer
36 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
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0answers
44 views

Series expansion of inverse polynomial

Suppose an nth order polynomial $P_n(x)$ with real and distict 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
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1answer
39 views

Sum of a series with exponential and polynomial terms

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
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2answers
50 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 ...
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2answers
41 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
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1answer
38 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 ...
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1answer
188 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^...
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1answer
70 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?
2
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0answers
46 views

Series expansion of elliptic integral involving n th order polynomial in the denominator

My goal is to find an expansion in powers of 1/ρ of the integral: \begin{equation}I_n(\rho)=\int_\rho^{+\infty}\frac{dt}{(E_n(t))^2\sqrt{t^2-h_2^2}\sqrt{t^2-h_3^2}},\quad \rho \ge h_2\end{equation} ...
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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
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1answer
62 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
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1answer
34 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?
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1answer
37 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 ...
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1answer
119 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
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1answer
174 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. $$\...
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0answers
31 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 $w=i$ ...
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0answers
35 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 $\phi$ ...
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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
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1answer
19 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
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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 $$\sum_{n=1}^{\infty}n^2 \frac{x}{n^4}$$...
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2answers
58 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
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1answer
65 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
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2answers
133 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 \...
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3answers
66 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+\...
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1answer
34 views

Radius of convergence and the existence of antiderivative

I think I have some misunderstandings regarding some basic concepts. First, the question I'm dealing with is the following: Let $f$ be analytic in $\{z ;|z|>1 \}$, and $\int_{|z|=2}f(z)dz=0$. ...
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1answer
40 views

Multidimensional taylor series $sin (x^3y^2) $

A homework of mine was to compute the Taylor series of $f(x,y)=\sin(x^3y^2)$ around $(0,0)$ to the 25th order. I assumed, as $\sin(z)=\sum\limits^{\infty}_{k=0}(-1)^k\frac{z^{2k+1}}{(2k+1)!}$, that I ...
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1answer
42 views

Series expansion of $1/(1+z^2)$ about the point I

I am trying to find a series representation for the complex function: $1/(1+z^2)$. The text I am reading gives: $1/(1+z^2) = 1/((z+I)(z-I)) = -I/(2(z-I)) +1/4 - I(z-I)/8 - (z-I)^2/16 + ...$ I do not ...
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0answers
71 views

Maclaurin Expansion of $\ln(3+x)$

I'm currently evaluating a simple Maclaurin expansion, the confusion I have with is why the expansion of this function is constructed to be: $\ln\left[3\left(1+\dfrac{x}{3}\right)\right]$ as opposed ...
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3answers
65 views

What is the power series expansion at $x=0$ of the algebraic function defined by $(27x-4)y^3 + 3y + 1 = 0$?

Let $y$ denote the complex-valued algebraic function defined implicitly near $x=0$ by $(27x - 4)y^3 + 3y + 1=0$ and such that $y(0)=1$. What is the power series expansion of this function at $x=0$? ...
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1answer
39 views

Proof of a formula containing double factorial

How can I prove the formula: $$\sum_{k=0}^\infty\dfrac{x^k}{k!!}=\dfrac{1}{2}e^{\dfrac{x^2}{2}}\left[2+\sqrt{2\pi}erf\left(\dfrac{x}{\sqrt\pi}\right)\right]?$$ Thanks
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3answers
39 views

On the composition of formal power series

I an attempt to compute the coefficients of the composition $f(g(x))$ of two power series $f(x) = \frac{1}{1-x}$ and $g(x) = \frac{1}{1-x}-1$, I used the definition of composition to get to $$f(g(x)) ...
4
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6answers
137 views

Deriving power series for $\sin x$ without using Taylor's Theorem or $\exp z$

Starting with defining $(\cos t, \sin t)$ from the unit circle, is it possible to derive the power series for $\sin(t)$: $$\sin t = t - \frac{t^3}{3!} + \frac{t^5}{5!} - \dots$$ Note: I will be ...
2
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1answer
121 views

Prove $\sqrt{1+x}$ can be represented by a power series

I need to show that $\sqrt{1+x}$ can be represented as a power series. I need to prove the equality between the function and its Taylor series, not to prove that the Taylor series of the function, $\...
1
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1answer
30 views

Differentiated series of a power series has the same radius of convergence

I am trying to prove that the radius of convergence of a power series does not change after differentiating term by term. Let $\sum a_nx^n$ be a power series with radius of convergence $R$. Let $R_2$ ...
3
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3answers
457 views

Interesting Series with Zeta Function

I was trying to find another representation for the value of an integral when I found the following series: $$f (z)=\sum_{n \in \Bbb N} (-z)^{n-1}\frac {(2^n-1)}{2^n}\zeta (n+1) $$ For $|z|<1$ and ...
0
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1answer
30 views

Creating Formula from Data Series

I have a power-law-ish data series and need to back my way into a formula for it, such that it can be generated for any number of points. This is the n=20 version (value rounded to nearest 0.25). ...
1
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1answer
44 views

Geometric proof of expansions of series

I have read that Barrow had proved the fundamental theorem of calculus. I have read that proof and its a good. Further I know Newton had derived the sine and cosine series. His methods obviously didn'...