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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2answers
28 views

Transformation of a function into a power series

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
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1answer
25 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
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0answers
21 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
21 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
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2answers
70 views

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

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?
3
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4answers
168 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
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2answers
38 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?
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0answers
15 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
28 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
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0answers
26 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
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2answers
87 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
467 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
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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
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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
18 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 $\epsilon > 0$?
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0answers
44 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
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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
38 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
50 views

How can I solve the following differential equation [on hold]

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
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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
58 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
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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
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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
17 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
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1answer
21 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
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1answer
54 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
37 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
593 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
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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 ...
1
vote
2answers
45 views

Find the function that equals to $1-x^3+x^6-x^9+ \cdots$

Find the function that equals to $1-x^3+x^6-x^9+ \cdots$ for all $|x| < 1$ I know that $\frac{1}{1+x} = 1-x+x^2-x^3+...$ But I couldn't find the pattern here
1
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1answer
32 views

how to understand Taylor's inequality intuitively?

I am learning the Taylor Series at the moment and I am trying to figure out how to understand Taylor's inequality intuitively. I know you can integrate repeatedly and prove the inequality is ...
0
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1answer
311 views

Difference between power series method and Frobenius method

There is the power series method for solving ordinary differential equations: one looks for solutions of the form $\sum c_n x^n$, and derives algebraic relations between coefficients $c_n$. Then ...
0
votes
3answers
23 views

Power series expansion using Taylors Theorem.

So the function $f(x)=3x^2-6x+5$ needs to be written as a power series expansion around $x=a$ and the goal is to show $x=a$ is $f(x)$ for every $a$. So I started off by finding up to the third ...
0
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0answers
19 views

Help with simplification rules form sums and integrals.

IF you had a power series with summation notation and an integral what expressions would you be able to pull outside the integral and which would you be able to pull outside the sum.
0
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1answer
20 views

Help with general power series concept

If f(x) is some general polynomial, what will the power series expansion of f(x) be. Is there a set rule for finding the power series of polynomials.
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2answers
59 views

How can I solve the following differential equation without use power series [on hold]

Let we have the following differential equation $$y''-xy'=e^{-x}$$ how can I solve this differential equation without use power series
0
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0answers
11 views

Recursive relationship for Peano Baker Series

The Peano Baker Series is a integral has the following form $$\varPhi(h,0)=I+\intop_0^h G(t_{1}) \, dt_1 + \intop_0^h G(t_1) \intop_0^{t_{1}} G(t_2) \, dt_2 \, dt_1 + \intop_0^h G(t_1) ...
-1
votes
2answers
32 views

How can I solve the following differential equation [on hold]

How can I solve the following differential equation by power series near the point $z=1$ $$(z^2-2z+2)w''+2(z-1)w'=0$$ Then I have to find the radius of convergence of the solution
1
vote
3answers
36 views

How to solve power series expansions.

The function is $f(x)=1/(1-x)$ and it asks to find a power series expansion expanded around $x=a$, which would be the general expansion as well as around $x=0$ and $x=2$.
1
vote
1answer
42 views

How do I extrapolate summation notation from a given series?

I am currently working on the power series for a homework assignment. I have to find the radius of convergence for the function $$\frac{10}{1+64x^2}$$ By setting up the $$\frac{1}{1+64x^2}$$ part ...
0
votes
1answer
32 views

An entire function is a polynomial iff the Taylor expansion around $0$ converges uniformly

Let $g:\mathbb{C} \to \mathbb{C}$ an entire function. Prove that the Taylor expansion around $0$ converges uniformly in all $\mathbb{C}$ if and only if $g$ is a polynomial. 1/2 PROOF I think I ...
0
votes
2answers
19 views

Estimate on rate of growth of a power series

Given two sequences $(a_k),(b_k)$ with $a_k\geq0,b_k>0$ such that the power series $\sum_{k=0}^\infty a_k b_kr^{k}$ and $\sum_{k=0}^\infty a_kr^k$ converge for each $r>0$. My question now is: ...
-2
votes
0answers
17 views

How to solve power series for terms and convergence?

Find the first five non-zero terms of power series representation centered at x=0 for the function. What is the interval of convergence? f(x) = \frac{x}{1+3 x}
1
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1answer
15 views

Multiplying and factoring in Formal Power Series

I'm working with some formal power series in my homework. Somewhere in the middle of my hw problem I reach a point where I would really like to factor, but I'm not sure if I can. Suppose $F_k$ ...
2
votes
2answers
72 views

Prove $f(x)=g(x)$ for all $x \in\mathbb{R}$

If $$f(x)=\sum_{n=0}^\infty\frac{x^n}{n!}, x\in\mathbb{R}$$ and $$ g(x) = 1 + \int_0^x f(t) \,dt $$ prove that $g(x)=f(x)$ for all $x\in\mathbb{R}$ and prove that $f$ is differentiable on ...
0
votes
1answer
344 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 ...