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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0
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1answer
727 views

Y-Axis units on FFT graph

A 50Hz sinusoid wave with a voltage range of +/-20V is sampled at 512Hz for 1 second. No bias or phase shift are present. The signal is run through an FFT. The result is one spike at 50Hz on the ...
1
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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
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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
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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 $x$.
5
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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 ...
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 $$\sum_{n=1}^{\infty}n^2 \frac{x}{n^4}$$...
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 ...
5
votes
1answer
165 views

Infinite series $\sum_{n=1}^{+\infty} a_n x^n$ is convergent when $x=-3$, and it is divergent when $x=20$

Assume that the series $\displaystyle\sum_{n=1}^{+\infty} a_n x^n$ is convergent when $x=-3$, and it is divergent when $x=20$ Is the series $\displaystyle\sum_{n=1}^{+\infty} a_n (-2)^n$ divergent? ...
-1
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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+\...
1
vote
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 $\int_{|z|=2}f(z)dz=0$. ...
0
votes
1answer
44 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 ...
2
votes
1answer
42 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 ...
1
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0answers
73 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 ...
1
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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 $x=0$? ...
2
votes
1answer
122 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, $\...
0
votes
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
4
votes
6answers
140 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
votes
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)) ...
1
vote
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
460 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 ...
4
votes
3answers
141 views

There's a small detail in this proof on why $\sum_{k=1}^{\infty}\frac{1}{k^2} = \frac{\pi^2}{6}$ that I can't figure out

http://www.maa.org/sites/default/files/pdf/upload_library/2/Kalman-2013.pdf Here is a link to the article I have been reading. It's really interesting and easy to follow. What bothers me is a result ...
0
votes
1answer
36 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
47 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'...
0
votes
1answer
35 views

Logarithm on the reals: negative power series?

I am well aware that $ln(x)$ has no Laurent series on the complex plane, because of its multi-valuedness, but I have always wondered whether a similar beast would exist if we restricted our attention ...
2
votes
0answers
45 views

Radius of convergence of power series $\sum\limits_{n=1}^{\infty}n!x^{n!}$ (different methods yield different results)

I have to find the radius of convergence of $\sum\limits_{n=1}^{\infty}n!x^{n!}$. It is a power series, therefore: One of the ways to find the radius of convergence is to find $\lim \dfrac{|n!|}{|(...
1
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1answer
34 views

Unclear definition concerning convergence of a power series

I am following a course on computer algebra and at the end of the course, my professor wrote something down I could not follow at first. Concretely it handles about a definition given as follows: ...
1
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1answer
46 views

Taylor series for $\tanh(z)$

Find the Taylor series of $\tanh(z)$ around $z_0=0$. $z$ is a complex variable. I can use all the basic series as facts like the $\cosh$ and $\sinh$ series. I know how to calculate the series ...
1
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0answers
29 views

Convergence of $\sum_{n=1}^{\infty}\log(1-e^{2\pi inz})$

Let $f(z)=\sum_{n=1}^{\infty}\log(1-e^{2\pi inz})$ be given on the upper half plane $H=\{z\in\mathbb{C}:\Im(z)>0\}$. Why does this function converges absolutely and uniformly on compact subsets of $...
0
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0answers
25 views

Branch points of power series and radius of convergence

The following text trys to give an answer to the following problem: Let $t(q)=\sum_{n=0}^{\infty}t_n q^n$ be a powerseries with radius of convergence $1$ and $t_0=0,~t_1\neq 0$. $w(q)$ is another ...
1
vote
1answer
16 views

Convergence of factor

In my math course there are some examples to test convergence of power-series with d'alembert. One of the examples is : $\sum_{n=0}^\infty \frac{x^n}{n!}$ Now i tried to solve this with d'...
1
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1answer
22 views

Convergence of sum of power series and numerical series.

Considering the following series $$\sum_{n=1}^{\infty} \frac{n}{2^n}(x+2)^n + \sum_{n=1}^{\infty} \frac{n^3}{\sqrt {n!}}$$ We need to calculate the domain of convergence of this series. Well the ...
3
votes
2answers
272 views

Is it true that $log(i) = \frac\pi2i$ ? If so, are both of these legitimate proofs? They seem too beautiful not to be…

Sorry if this is a naive question. I have not yet taken any upper level math courses involving complex numbers. However, in preparation for those courses, together with utilizing the knowledge that ...
1
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1answer
27 views

Radius of convergence

What is radius of convergence of $\sum_{n=2}^{\infty} \left(1+\dfrac{1}{n}\right)^{n^2}z^n$? I don't know anything
0
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0answers
58 views

Is it possible to find $g(\kappa)$ in this equation

I have ran into the following integral equation as part of my research. For $\xi = (\alpha\theta)^{1/\alpha}$ and for all $\theta>0$. I have the following equality $$\int\limits_0^\infty g(\...
1
vote
2answers
108 views

If $\sum_{n=0}^{\infty}a_{n}x^n $ converges for $|x| < R$ , then $\sum_{n=0}^{\infty}na_{n}x^n $ converges for $|x| < R$

I have the following statement: If $\sum_{n=0}^{\infty}a_{n}x^n $ converges for $|x| < R$ , then $\sum_{n=0}^{\infty}na_{n}x^n $ converges for $|x| < R$ as well. I couldn't find a ...
1
vote
1answer
43 views

Calculus 2 - Prove Disprove - convergence of Taylor series

I got this question regarding properties of Taylor series. I'm stuck on the second question, I believe it is true since the area of convergence for X is affected by the coefficient and it is not ...
1
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2answers
37 views

Expanding a function into a power series

I was given this function $$\frac{x-2}{1-x}$$ around x zero = 2 I'm not sure how to do it, i called x-2 = t, so 1-x = -t-1 and i moved on from there.. Is there another way to do this question?
1
vote
1answer
15 views

About the set of $x$ values at which the Taylor series of $f(x)$ converges to $f(x)$

Let $f(x)$ be a function (for simplicity, let us assume that it is defined on $\mathbb{R}$ and infinitely differentiable), and $T$ the Taylor series of $f$ at $x=a$, with interval of convergence $I$. ...
6
votes
0answers
110 views

Help with the following summation when $x^{37}=1,x\neq 1$

I want to find the following summation $\text{Let }x^{37} = 1 \text{ and } x \neq 1,$ $\\ \text{Find the summation of }$ $$\frac{1}{(1+x+x^2+x^3)^3}+\frac{2}{(1+x^2+x^4+x^6)^3}+...+\frac{36}{(1+x^{...
0
votes
2answers
80 views

Explain why $\sum\limits_{k=0}^\infty k^2\cdot x^k=-\frac{x(x+1)}{(x-1)^3}$

I was learning generating functions and met this summation. I used maple and it gave $-\frac{x(x+1)}{(x-1)^3}$, but how does it get here? I've forgotten most of the knowledge about series. Does anyone ...
7
votes
2answers
1k views

What is the radius of convergence of $\sum z^{n!}$?

How to find the radius of convergence of $\sum z^{n!}$? I'm used to applying the ratio test to power series of the form $\sum a_{n}z^{n}$, but for a different power of $z$, I am a bit stumped. What ...
-4
votes
2answers
114 views

Identifying $\sum\limits_{k=0}^\infty\frac{1}{k!}\frac{x^{k+6}}{k+6}$ [closed]

I'm trying to prove this equality. $$\sum_{k=0}^\infty\frac{1}{k!}\frac{x^{k+6}}{(k+6)} {=} e^x(x^5-5x^4+20x^3-60x^2+120x-120)+120$$ posted by: http://math.stackexchange.com/q/832368. How do I get ...
0
votes
2answers
52 views

Explicit expression of a given power series

Let us have a look to the power series of the form $$\sum_{n=0}^{\infty}{\frac{1}{n+2}x^n},\ \ \ x\in\mathbb{R}$$ I want to find an explicit expression of this power series. I think one have to us ...
1
vote
1answer
36 views

Radius of convergence of $\sum_{k=0}^{\infty} c_n^2x^n$.

The problem statement is as follows: Suppose the radius of convergence of the complex valued series $\sum_{n=0}^{\infty} c_nz^n$ is $R$. Find the radius of convergence of $\sum_{n=0}^{\infty} c_n^2z^n$...
1
vote
1answer
14 views

Help to prove an expression about sums of binomials coefficients using Complex Power Series theorem.

I'm solving some exercises from Kreszig's Advanced Math book and I got stuck in one: (10th ed, chapter 15.3, problem 18): Using $(1+z)^p*(1+z)^q=(1+z)^{p+q}$, obtain the basic relation: $$\sum_{n=0}^...
0
votes
2answers
31 views

Radius of convergence of a series which diverges when every term is made positive

$\{\,a_n\mid n \ge 1\,\}$ be sequence of real numbers. Partial sum of $a_n$ forms a convergent series. Partial sum of absolute value of $a_n$ forms a divergent series. Let radius of convergence of ...
0
votes
1answer
23 views

Series Solution For ODE

I am currently working on some introductory problems for series solutions for ODEs and am really struggling. The question is as follows: $$ (7+x)y' = y $$ Calculate the first five terms in the series....
2
votes
2answers
90 views

radius of convergence of Taylor series, function with branch cuts

Let $f(z) $ being the analytic continuation of some holomorphic function, having many branch points and isolated singularities at $\beta_1,\beta_2,\ldots,\beta_n,\ldots$ is the radius of convergence ...
0
votes
0answers
25 views

How to prove a self-recirpocal polynomials $P(z)$ to have all its zeros on the unit circle $|z|=1$?

Let $m(n)=10(n+1)^3$ and $$c_j(n)=\frac{2 (2j+1)}{\Gamma(j)}\sum_{k=1}^{n}(\pi k^2)^{j}\tag{1}$$, $$P(z)=\sum_{j=1}^{m(n)}(-1)^jc_j(n)\left(z^{4j+1}+z^{-(4j+1)}\right)\tag{2}$$ $$Q(z)=z^{4m(n)+1}P(z)=...