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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2
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1answer
19 views

Changing order of summation including a min in the summation

Lets say I have the following expression: $$ h(x) = \sum_{k=1}^n \sum_{v=1}^{\min\{k,j\}} \frac{(-1)^{n-k}k!}{(k-v)!} {n \brack k}f(x)^{k-v} B_{n,v}^f(x) $$ Now my goal is to have the $v$ ...
3
votes
3answers
932 views

General term of Taylor Series of $\sin x$ centered at $\pi/4$

What is the general term for a Taylor series of $\sin(x)$ centered at $\pi/4$? It should be $(-1)^{[??]} \times \sqrt{2}/2 \times \frac{(x-\pi/4)^n}{n!}$ What power is $(-1)$ supposed to be raised ...
3
votes
0answers
32 views

$\pi$ base $e$ or $\pi=\sum\limits_{n=-1}^{\infty} a_ne^{-n}$ where $a_n\in\{0,1,-1\}$

I was "playing with $\pi$" trying to look at it in different numeral systems and it's not so hard to obtain $\pi$ base $2$ or $3$ or even $\varphi=\frac{\sqrt{5}+1}{2}$, using Maclaurin series of ...
1
vote
0answers
16 views

$\lim\sum_{k=0}^{\lfloor\delta n\rfloor} \frac{n^k}{k!}e^{-n}$ and Poisson distribution

Problem: Let $X_1,X_2\ldots$ be some independent random variables with Poisson distribution with parameter 1. Show that for every $\epsilon > 0$ sequence $S_n-(1-\epsilon)n$ converges to ...
0
votes
1answer
39 views

How would I set up the Taylor's Inequality to prove that the function is equal to its Taylor Series expansion?

How would I set up the Taylor's Inequality to prove that the function $f(x) = \frac{1}{x}$ is equal to its Taylor Series expansion centered at $x=1$? I've done the Taylor series expansion, but ...
2
votes
4answers
100 views

Write a function as $\sum _{n=0} ^{\infty} a_n x^n$

We have $f(x) = (x+ x^2 + x^3 + x^4 + x^5 + x^6)^4$. Now I want to write this as $\sum _{n=0} ^{\infty} a_n x^n$. What I got: $f(x) = (x+ x^2 + x^3 + x^4 + x^5 + x^6)^4 = x^4 (1+ x + x^2 + x^3 + ...
0
votes
1answer
27 views

One of the points of interval of convergence of the power series $\sum_{n=0}^\infty \left(\frac{x^8-1}{3}\right)^n$ is not a real number?

One of my points is not a real number in my interval of convergence! This is how I calculated it: $$\lim_{n\to\infty} \left|\frac{x^8-1}{3}\right|$$ Using root test $\frac{x^8-1}{3} <1$ ...
-1
votes
0answers
25 views

How can I solve differential equation near point that is not normal

Let we have the following differential equation : $$2z(z+1)w''+z(z+1)w'-w=0$$ By power series near the point $z_0=0$ the problem that the point $z_0$ isn't normal point for this equation , so how can ...
3
votes
6answers
193 views

How to Find the Function of a Given Power Series?

(Please see edit below; I originally asked how to find a power series expansion of a given function, but I now wanted to know how to do the reverse case.) Can someone please explain how to find the ...
0
votes
2answers
27 views

Find a recurrence relation and solve it

Let $a_n$ be the nummber of ways that 4 people can throw $n$ eyes together with a die. Every person throw once. Now I want to find a generating function and compute $a_n$ for different $n$. To do ...
1
vote
1answer
19 views

Finding interval of convergence for complicated sum

I'm going through old exams for my Calc III course and came across a problem that I did not know how to do. The problem is: Find the interval of convergence of the series ...
1
vote
3answers
59 views

Compute the sum $\sum_{k=1}^{\infty}k^mz^k$ where $|z|<1$

Do you know how to find the limit of $\sum_{k=1}^{\infty}k^mz^k$ where $|z|<1$ and m is a natural number? I've tried to google it in wiki but I do not understand the closed form ...
0
votes
1answer
29 views

Stuck on finding where $\sum_1^{\infty} (x+4)^{n}$ converges conditionally.

I have all the questions correct on my hw except for one: find where $\sum_1^{\infty} (x+4)^{n}$ converges conditionally. Radius of Convergence I got 1 for this, by using the root test and finding ...
1
vote
1answer
47 views

Is a holomorphic function analytic in a ‘real’ sense?

I am taking a course in complex analysis, and I asked myself the following question: If a function $ f: \mathbb{C} \to \mathbb{C} $ is holomorphic, can its real and imaginary parts be given by a ...
0
votes
1answer
32 views

Find series representation of a function

Let $\displaystyle f(x) = x^4 (\frac{1-x^6}{1-x})^4$. Now I want to find the series representation of this function. Note that $\displaystyle \frac{d^3}{dx^3} (\frac{1}{1-x})= \frac{6}{(1-x)^4}$. ...
0
votes
2answers
32 views

Calculating a power series [on hold]

I was wondering if anyone knows how to calculate: $\sum_\limits{t=-\infty}^{\infty}$ $a^{t} e^{-itb}$, for constants a,b and $-\pi < b < \pi$ Can we take the t=0 term out to reduce it ...
2
votes
0answers
30 views

Power series writing terms

Can someone explain how I should be solving for theses terms? I got my series to be $10(-7x)^n$, but I dont know what to do from there. Original link to image: http://imgur.com/6sUVvaR
1
vote
2answers
38 views

Transformation of a function into a power series [on hold]

How can I transform the real functions $\frac{1}{1-\sin(x)}$ and $\frac{x}{e^x-1}$ into power series with $x_0=0$?
1
vote
1answer
34 views

Poles of power series

This may be a trivial question, but I haven't been able to find an answer. Given a power series about $x_0$ $F(x)=\sum_{n=0}^\infty a_n (x-x_0)^n$, how do we find its (complex) poles? What about the ...
2
votes
0answers
25 views

Formal Expansion of another Expansion

Given a function $f(x)=\sum_{n=1}^{\infty}\frac{c_n}{x^n n!}$, where $c_n$ are constants, we want to find the formal series expansion of the function $g(x)=\exp(f(x))$ in terms of $x$. I want to ...
1
vote
1answer
29 views

Power Series: $\sum_1^{\infty} (x)^{n}\frac{n^3}{n!}$

I just started learning about the power series, can someone help me with finding the radius of convergence and interval of convergence? So I am stuck on the radius of convergence because apparently I ...
0
votes
2answers
72 views

How to find a Taylor series for $e^{x^2-1}$? [closed]

How do I proceed to write a taylor series expansion for $e^{x^2-1}$? I know the series for $e^x$: it is $1+(x)+(x^2/2!)+\dots$ Edit: Would a Maclaurin series expansion be different?
4
votes
4answers
177 views

The even-numbered coefficients of the Maclaurin series of $ \frac{1}{\cos(x)} $ are odd integers.

Let’s consider $ G(z) \stackrel{\text{df}}{=} \dfrac{1}{\cos(z)} $ as the exponential generating function of the sequence of Euler numbers. How can one prove that in the Maclaurin series of $ G $, $$ ...
0
votes
2answers
45 views

Function represented by power series

To what function does the function with power series , $ |x|<1$ $$F(x)=\frac{x^2}{2}-\frac{x^4}{4}+\frac{x^6}{6}-\frac{x^8}{8}+\cdots$$ converge?
0
votes
0answers
16 views

Series representation of simple function - a general form for the coefficients?

I'm looking for a series representation for $$ f\left(r_j\right)=\frac{ \left( m - r_j \right)^{\frac{3}{2}\left(m-1\right)}}{\left(j + m - r_i - r_j \right)^{\frac{3}{2} \left( m + j - 1 \right)} } ...
1
vote
5answers
31 views

Infinite sequence and power series

infinite sequence $a_{n}$ where $$\lim_{n\to \infty} |na_{n}|=1101 $$ Find R of convergence of the power series $$\sum_{n=1}^\infty a_{n}x^n$$ Anyone can guide me for this question? Thank you so ...
0
votes
0answers
30 views

Compute radius of convergence and the first three coefficients of a function

Let $\displaystyle f(z) = \frac{z+1}{(2z+1)(1+ \sin z)}$, with serie expansion $\sum_{n=0} ^\infty a_n z^n$ around zero. Now I want to compute the radius of convergence and the first three ...
1
vote
2answers
93 views

Representation of power series of product of sine and cosine

Given $$ f(x)= \int \limits_0^x \sin(y^2) \cos(y^2) \mathrm{d}y $$ Anyone can help and guide me for this?I don't really have an idea of how to represent it as power series Thank you! My attempt: $$ ...
7
votes
5answers
477 views

Why the radius of convergence and not “areas of convergence” for power series?

My calculus is quite rusty and I'm trying to rebuild it on an intuitive basis. Currently, I am looking at power series and have trouble understanding the radius of convergence. I am comfortable with ...
0
votes
0answers
33 views

Radius of convergence of the series-power series

Can anyone help me to check whether my solution is correct because we are not provided with the solutions,but I want to ensure what I did is correct. Thanks for your help! (a)$\sum_{n=1}^\infty 5^n ...
1
vote
1answer
27 views

Determine the radius of convergence of $\sum_{n=1}^\infty n^{n^{1/3}}z^n$ (by the ratio test if possible)

Determine the radius of convergence of the following power series: $\sum_{n=1}^\infty n^{n^{1/3}}z^n$ Applying the ratio test gives $\frac{({n+1})^{({n+1})^{1/3}}}{n^{n^{1/3}}}z<1$. So ...
1
vote
1answer
19 views

Complex power series which converges absolutely on the boundary converges absolutely on a neighborhood of the boundary

If a complex power series $\sum_{n = 0}^{\infty} a_n z^n$ converges absolutely for $|z| \leq 1$, does it necessarily converge absolutely for $|z| < 1 + \epsilon$, for some $\epsilon > 0$?
0
votes
0answers
45 views

how would i simplify this into an identity?

$$ B_{n,k}^{f\ln(g)} = B_{n,k}\left(\frac{d}{dx}[f(x)\ln(g(x))], \frac{d^2}{dx^2}[f(x) \ln(g(x)), \cdots, \frac{d^{n-k+1}}{dx^{n-k+1}}[f(x) \ln(g(x))]\right) $$ We know that: $$ B_{n,k}^{f\ln(g)} = ...
1
vote
2answers
42 views

approximate $\int_{0}^{0.5}{\frac{\sin(x)}{x}}dx$

By using Maclaurin series, approximate the value of $$\int_{0}^{0.5}{\frac{\sin(x)}{x}}dx$$ to within an error $0.0001$, where $x$ is in radians. My attempt: Since we know the Maclaurin series of ...
0
votes
1answer
42 views

Radius of convergence and sum of alternating series $1 - z + z^2 - z^3 + \ldots $

I have a (complex) function represented by the power series \begin{equation*} L(z) = z -\frac{z^2}{2} + \frac{z^3}{3} - \frac{z^4}{4} \ldots \end{equation*} which I have tried to represent (perhaps ...
3
votes
1answer
56 views

Is my proof that $\frac{\pi}{4}=\sum\limits_{n\geq 0}(-1)^n \frac{1}{2n+1}$ correct?

Respected All I was trying to prove that $$\sum_{n\geq 0}(-1)^{n} \frac{1}{2n+1}=\frac{\pi}{4}$$ What I tried to show like this. We know $$\frac{1}{1+x^2}=(1+x^2)^{-1}=\sum_{n\geq 0}(-1)^nx^{2n}, ...
-1
votes
1answer
55 views

How can I solve the following differential equation [closed]

How can I solve the following differential equation : $$w''+(\sin z)w'+(1+z^2)w=0$$ In two case : without use power series use power series near the point $z=0$
1
vote
2answers
178 views

Uniform convergence of the series for $\frac{x}{\sqrt{1+x}} = \sum_{n=1}^\infty (-1)^n\binom{-1/2}{n}\left(\frac{x}{1+x}\right)^{n+1}$

I am looking for the values where this series expansion converges uniformly. $$\frac{x}{\sqrt{1+x}} = \sum_{n=1}^\infty (-1)^n\binom{-1/2}{n}\left(\frac{x}{1+x}\right)^{n+1}$$ Intuitively, I believe ...
1
vote
3answers
63 views

Working out $\tan x$ using sin and cos expansion

Using only the series expansions $\sin x = x- \dfrac{x^3} {3!} + \dfrac{x^5} {5!} + ...$ and $\cos x = 1 - \dfrac{x^2} {2!} + \dfrac{x^4}{4!} + ...$ Find the series expansions of the $\tan x$ ...
0
votes
0answers
23 views

power series steps help

Can someone show me how to do this problem? I really need a walk through of the steps if possible. Find a power series representation for the function and determine its radius of convergence. $$ ...
0
votes
0answers
64 views

How to solve $1 = \sum_{p \text{ prime}} x^{-p-1}$?

As the title says, I am trying to solve the equation $$1 = \sum_{p \text{ prime}} x^{-p-1}$$ and I'm not really sure where to begin. I got this from an exercise in a book and apparently there is a ...
0
votes
4answers
28 views

Power Series representation of $\frac{x^3}{(3x+4)^2}$

How do you do this? I have an exam in 2 hours and I know this type will be on it and I have no clue. We were taught to base it off the power series of $x^n$
0
votes
1answer
18 views

Difference between a convergent series and an asymptotic series?

Can someone let me know the difference between a convergent series and an asymptotic series with an example? Can both the series be the same at some situations? In what situations an asymptotic series ...
1
vote
1answer
22 views

Power series confusion

I'm having trouble with power series. Can someone walk me through this? My biggest problem is always figuring out what I need to break apart. Find a power series representation for each function ...
1
vote
1answer
55 views

Infinite series $\sum_{n=1}^{\infty}nx^{n+1}$ does not comply to any of my (known) tests

I am attempting to find the interval of convergence for $$\sum_{n=1}^{\infty}nx^{n+1}$$ The lower bound, x = -1, would be tested by determining if $$\sum_{n=1}^{\infty}n(-1)^{n+1}$$ diverges. ...
5
votes
1answer
38 views

Debye Function Integral (BlackBody)

Show that $$ \int^{\infty}_{0} \frac{x^{3} \, dx}{e^{x}-1} = \frac{\pi^{4}}{15} $$ by expanding the integrand in powers of $e^{-x} $ and integrating term by term. Could anyone help with this one?
0
votes
1answer
22 views

Power series expansion of f(x)=1/(1-x) around x=0 and x=-1

For the power series expansion of the function $f(x)$ I worked out the at $x=0$ the power series expansion is $$1(x-0)^n$$ and at $x=-1$ the power series expansion is ...
6
votes
4answers
603 views

If $\sum_{n = 1}^\infty {{a_n}}$ converges, then is $\sum_{n = 1}^\infty (1+a_n)^{-1}$ a convergent series?

If $\sum\limits_{n = 1}^\infty {{a_n}}$ is convergent (with ${a_n} > 0$, $\forall n\in\mathbb{Z}$), then is $\sum\limits_{n = 1}^\infty {\left( {\frac{1}{{1 + {a_n}}}} \right)}$ is a convergent ...
1
vote
2answers
53 views

radius of convergence of $\sum_{n=1}^\infty n!^2x^{n^2}$ [closed]

Determine the radius of convergence of the following power series: $\sum_{n=1}^\infty n!^2x^{n^2}$
1
vote
0answers
23 views

Radius of convergence of $x/sinh(x)$

the function $\mathbb{R}\ni x\mapsto \frac{x}{\sinh(x)}\in\mathbb{R}$ can be written in a neighborhood of $0\in\mathbb{R}$ as a Taylor series, i.e. $\frac{x}{\sinh(x)}=\sum\limits_{k=0}^{\infty} a_k ...