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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-4
votes
1answer
75 views

Find the value of Taylor coefficient $a_5$ for the function $ \int_0^x (e^{-t^2}+\cos t) \, dt$

Find the value for $a_5$ If $ \int_0^x (e^{-t^2}+\cos t) \, dt$ has the power series expansion $\sum_1^\infty a_nx^n$, then find $a_5$ up to three correct decimal places. I think it is a Taylor ...
1
vote
1answer
70 views

Find the Laurent expansion for $f(z)=\frac{\exp{1/z^2}}{z-1}$ about $z=0$.

Find the Laurent expansion for $f(z)=\frac{\exp{(1/z^2)}}{z-1}$ about $z=0$. I was able to determine the series for each of the factors. We have ...
2
votes
1answer
40 views

Maclaurin Series: Complex Analysis

Question: Use the representation $\sin z = \sum\limits_{n=0}^\infty (-1)^n\frac{z^{2n+1}}{(2n+1)!}$, $|z|<\infty$ to write the Maclaurin series for the function $f(z) = \sin z^2$ and point out how ...
3
votes
1answer
71 views

Find the Laurent Expansion of $f(z)$

Find the Laurent Expansion for $$f(z)=\frac{1}{z^4+z^2}$$ about $z=0$. I have found the partial fraction decomposition $$f(z)=\frac{1}{z^4+z^2}=\frac{1}{z^2}-\frac{1}{2i(z-i)}+\frac{1}{2i(z+i)}.$$ ...
0
votes
2answers
44 views

Proof for multiplication of two power series

Prove that $(\sum_{k=0}^\infty u^k)^2=\sum_{k=0}^\infty (k+1)u^k$ when |u|<1. This is a proof I need for a larger proof I was doing. I am stuck on this, so I was not able to make any notable ...
3
votes
1answer
68 views

Irreducibility in $k((t))[y]$

Let $k$ be an algebraically closed field of char $0$ and suppose $f(y) \in k[y]$ (need not be monic). Let $t$ be an indeterminate and consider the fraction field $k((t))$ of power series ring ...
1
vote
1answer
112 views

Real Analysis II — power series

If f(x)= $\int_0^x \frac {ln(1+u)}{u}$ du, find a series for f(x) and calculate the approximate value of f(0.1). Use the error upper bound for approximating an alternating series to give an upper ...
0
votes
1answer
48 views

Determine convergence of power series of triple integral

Does this power series converge? I think it does, but how to prove it? $$ \sum_{i=0}^\infty \int \int \int 3^{-i}\left(\cos(\pi x) + \cos(\pi y) + \cos(\pi z) \right)^idxdydz, $$ where the ...
0
votes
2answers
25 views

Power series representation of xln(3x+5)

I get to the point $\sum_{n=0}^{\infty }\frac{(-1)^n3^{n+1}x^{n+2}}{(n+1)5^{n+1}}$ by using the geometric series and integrating etc. But I looked up the answer and it is what I have plus the term ...
4
votes
3answers
98 views

Proving positivity of the exponential function

Question. Without using the semigroup property ($\mathrm{e}^{x}\mathrm{e}^{y}=\mathrm{e}^{x+y}$), how can we show that $\mathrm{e}^{x}>0$ for all $x\in\mathbb{R}$ only by using the series ...
5
votes
1answer
151 views

Power series expansion of Blaschke product

Suppose $B$ is a Blaschke product with at least one zero off the origin, and $B(z)=\sum_{k=0}^\infty {c_kz^k}$. Is it possible that $c_k\ge0$ for all $k=0,1,\ldots$? My try: Since $B(z)$ takes real ...
-1
votes
1answer
30 views

power series where $f(z)=e^z$ and $z_0=1$

How do i find the power series of the form: $$\sum_{n=0}^{\infty}a_n ({z-z_0)}$$ where $f(z)=e^z$ and $z_0=1$ using the geomatric series currently i have that it equals $$\sum_{n=0}^{\infty} ...
2
votes
2answers
55 views

What method was used here to expand $\ln(z)$?

On Wikipedia's entry for bilinear transform, there is this formula: \begin{align} s &= \frac{1}{T} \ln(z) \\[6pt] &= \frac{2}{T} \left[\frac{z-1}{z+1} + \frac{1}{3} \left( \frac{z-1}{z+1} ...
0
votes
0answers
18 views

Power series expansion of a transfer matrix in Matlab

As an example: $$H(z)=\left[\begin{matrix}\frac{3}{z+3} & \frac{5}{3z+3} \\ \frac{3}{z+4} & \frac{5}{2z+1} \end{matrix}\right]$$ How to use matlab to write $H(z) = ...
0
votes
1answer
112 views

Given a power series with interval of convergence $(-1,1]$, construct a series with another given interval of convergence

Suppose that you have a power series $$\sum_{n=1}^\infty (a_nx^n)$$ whose interval of convergence is $(-1,1]$. A) Using the same numbers $(a_n)$, come up with a new power series whose interval of ...
0
votes
0answers
32 views

Why does the limit of the p-series converge to a non-zero constant?

Here, the limit of the integral apparently converges to infinity as x goes to infinity. What I don't understand though, is how could it possibly converge to a constant of 5/7? If you multiply 5/7 ...
0
votes
0answers
14 views

Radius of convergence, correctly calculated?

I'm solving the proposed exercises in this PDF: http://math.bard.edu/belk/math142af09/ConvergencePowerSeries.pdf I solved the first exercise: How I can test if the radius I found (...
1
vote
1answer
40 views

Series solution

Given the differential equation $2(1-x)y''-3y'+\frac{y}{x}=0$ and in standard form: $y''-\frac{3}{2(1-x)}y'+\frac{1}{2x(1-x)}y=0$ I want to find the series solution for the larger root $σ = 1$ of ...
1
vote
2answers
147 views

Finding a recurrence relation, first few terms of power series solution to differential equation

I'm attempting to find a recurrence relation and the first few terms of a power series solution for the differential equation: $$(1-x^2)y \prime\prime - 2xy\prime + \lambda y = 0$$ Where $\lambda$ ...
2
votes
0answers
44 views

Finding a Taylor Expansion for the following:

I have: $$\frac{1}{1-z}$$ for $z_0=i$. I have no idea how to do the Taylor Series expansion of this, around $z_0=i$, and then show it summation form. I have: $\frac{1}{1-z} = ...
1
vote
1answer
40 views

General solution of $(1-x^2)y''-2y=0$ about $x_0=0$?

I've expanded this differential equation as a series to obtain the recurrence relation $$a_{n+2}=\frac{a_n(n^2-n+2)}{n^2+3n+2}.$$ I don't know how to find $a_n$ in terms of $a_0$ and $a_1$ so that I ...
1
vote
1answer
32 views

Abel's theorem for the derivative of a power series

Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a function, $(a_0, a_1, \dots)$ is a sequence of real numbers and $x_*$ is a positive real number, such that the following two conditions hold: for ...
0
votes
1answer
40 views

Power Series Question Relating to ArcTan

Why isn't the answer for this 9? If it is different how do I do it? Thanks in advance.!
2
votes
0answers
34 views

Closed form for $\sum_{k=1}^\infty \frac{k+1}{q^{k(k+n)}}$

What are, if they exist, closed forms for the series $$\sum_{k=1}^\infty \frac{1}{q^{k(k+n)}}, \qquad\qquad \sum_{k=1}^\infty \frac{k}{q^{k(k+n)}}, \qquad\qquad \text{ for }q>1,\;n\in \mathbb{N}?$$ ...
1
vote
3answers
41 views

Find Radius of Convergence of $\sum_{n=0}^\infty \frac{1}{2^n}\left(x-\pi\right)^n$

This is not a homework problem (I'm on break, so time for my own studies). Find the radius of convergence of \begin{align} \sum_{n=0}^\infty \frac{1}{2^n}\left(x-\pi\right)^n. \end{align} I have ...
0
votes
2answers
30 views

Proof of L'Hospital with power series

I'm having a bit of problem with this question. I feel like I have to prove the l'hospital's rule but I don't know where to start especially because I have to use the power series. Suppose that the ...
0
votes
3answers
38 views

is my radius of convergence correct?

Consider the series $\displaystyle \sum_{n=1}^\infty \left (1+{1 \over 2}+....+{1 \over n} \right )x^n$ . I applied the ratio test and I ended up getting it to simplify to: $$x \lim_{n \to \infty} ...
1
vote
1answer
44 views

The Taylor coefficients of a function of the form $\exp\circ f$, where $f$ is a power series

Let $(a_1, a_2, \dots) \in \mathbb{R}^\infty$ be a fixed sequence of real constants, and suppose the rule $$ x \mapsto \sum_{n = 1}^\infty a_n x^n $$ defines a function from the nonempty open interval ...
0
votes
3answers
41 views

Taylor Series for $\frac{1}{1+e^z}$ and radius of convergence

I have done some manipulation and got that $$\frac{1}{1+e^z} = \sum_{n=0}^\infty \frac{n!}{n!+z^n}$$ by the fact that: $$\frac{1}{1+e^z}= ...
0
votes
2answers
24 views

DTFT and its convergence

In the textbook "signals and systems", by prof. Simon Haykin, it says:   If $x[n]$ is not absolutely summable, but does satisfy square summable, then it can be shown that the following equation ...
0
votes
2answers
26 views

Partial Fractions as Power Series

I have the partial fraction sum $$f(i\omega)= a_0 + \frac{a_1}{\lambda_1+i\omega} + \frac{a_2}{\lambda_2+i\omega}$$ Which I want to represent as a power series in $ x = i\omega $ I thought that the ...
1
vote
1answer
25 views

Is radius of convergence correct for $\sum_{n=1}^{\infty} \frac{\bigg(3+(-1)^n\bigg)^n}{n}x^n$

I believe I have the correct answer but I'm not 100% confident in one of the simplification steps that I took: The series in question is $$F(x) = \sum_{n=1}^{\infty} ...
0
votes
3answers
54 views

Why does the series $\sum_{k=1}^\infty\frac{(1)^k}{k^2+k}$ converge?

In my homework for Differential Equations, we are determining interval of convergence for a given series. I've gotten the radius of convergence and found the unconfirmed interval of convergence ...
0
votes
0answers
23 views

Proof of identity with Hermite polynomials

Let's have Hermite polynomial, $H_{n} = e^{\frac{x^{2}}{2}}\frac{d^{n}}{dx^{n}}e^{-\frac{x^{2}}{2}}$. How to prove the identity $$ \tag 1 \sum_{n = 0}^{\infty}H_{n}(x)H_{n}(y)\frac{t^{n}}{n!} = (1 - ...
1
vote
2answers
48 views

Generating function for partitions

It is a theorem of Euler that $$\sum p(k)x^k=\prod\frac{1}{1-x^k}.$$ Something which annoys me is how to interpret the right hand side. I know that one can do this analytically, but I would like a ...
1
vote
0answers
20 views

How do I find a matrix for all power series solutions?

What is the “matrix” for $ \frac {d}{dx} $ acting on the vector space of all power series in the ordered basis $(1, x, x^2, x^3, ...)$? How can I use this matrix to find all power series solutions to ...
5
votes
4answers
55 views

Series of inverse function

$A(s) = \sum_{k>0}a_ks^k$ and $A(s)+A(s)^3=s$. I want calculate $a_5$. What ways to do it most efficiently?
1
vote
1answer
30 views

Uniform convergence of a complex power series on a compact set

I need to prove that the complex power series $\sum\limits_{n=0}^{\infty}a_nz^n$ converges uniformly on the compact disc $|z| \leq r|z_0|,$ assuming that the series converges for some $z_0 \neq 0.$ I ...
1
vote
1answer
34 views

Compute the 100th Bernstein polynomial for $e^x$

I need to find $$B_3 e^x = \sum_{k=0}^{100} e^{k/100}\binom{100}{k} x^k (1-x)^{100-k}$$ I can rearrange this to find $$\sum_{k=0}^\infty e^{k/100} \left(\frac{100!}{k!(100-k)!}\right) ...
0
votes
1answer
18 views

express tan(x) as a power series using maclauran's theorem. [duplicate]

the theorem states that if f(x) can be expanded as a power series for a given range of values of x then: $$f(x)=f(0)+xf'(0)+\frac{x^2}{2!}f''(0)+\frac{x^3}{3!}f'''(0)+\cdots$$ ($'$ means derivative) ...
5
votes
1answer
67 views

How to prove that the exponential function is the inverse of the natural logarithm by power series definition alone

The exponential function has the well-known power series representation/definition: $e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$ And the natural logarithm has the less well-known power series ...
1
vote
3answers
74 views

Radius of convergence of the power series $\sum_{n=1}^{\infty}a_nz^{n^2}$

Find the radius of convergence of the power series $$\sum_{n=1}^{\infty}a_nz^{n^2}$$ where , $a_0=1$ and $a_n=\frac{a_{n-1}}{3^n}$. My Work : We, have, ...
2
votes
2answers
25 views

Set of Convergence for the following Series

What is the set of convergence for this series: $ \sum_{n=1}^{+\infty} \dfrac{3^{\sqrt{n}}(2+i-3z)^n}{\sqrt{n^2+1}} $ ? My initial thought was to use, $ \dfrac{1}{R} = \lim(|a_n|)^{1/n}$, but this ...
0
votes
2answers
49 views

Power series of the solution of $2t^2x'' + tx' -(t+1)x=0$

Use the method of Frobenius, with the larger root of the indicial equation, to find the first three terms of the power series of the solution to $$2t^2x'' + tx' -(t+1)x=0.$$ My work: Note ...
0
votes
1answer
39 views

Radius of convergence of the power series $\sum x^{2^n}$

Find the radius of convergence of the power series $$\sum_{n=1}^{\infty}x^{2^n}.$$ Let , $u_n=x^{2^n}$. Then , $u_n^{1/n}=x^{\frac{2^n}{n}}$. Let, $m=\frac{2^n}{n}$ So, $m\to \infty$ as ...
2
votes
4answers
73 views

Find a power series for this function

$$f'(x) = 2xf(x) + 4x$$ I need to find the power series for $f(x)$, any hints on how this should be approached?
0
votes
2answers
31 views

Composition of real-analytic functions is real-analytic

Suppose $f,g: \mathbb{R} \to \mathbb{R}$ are real analytic, i.e, locally given by convergent power series. Then $g \circ f$ is real-analytic as well. How do I prove this? I guess the "standard" proof ...
1
vote
1answer
72 views

Find solution of $(1-x^2)y''-xy'+p^2y=0, p \in \mathbb{R}$

The following differential equation is given: $$(1-x^2)y''-xy'+p^2y=0, p \in \mathbb{R}$$ Find the general solution of the differential equation at the interval $(-1,1)$ (with the method of power ...
0
votes
0answers
28 views

Product of power series as a product of their coefficients

Suppose that $f(x)=\sum_{j=0}^\infty a_j x^j$ and $g(x)=\sum_{k=0}^\infty b_kx^k$ have positive radii of convergence $R_1$ and $R_2$ respectively. Let $c_n=\sum_{j=0}^n a_jb_{n-j}$ for $n\ge0$; and ...
3
votes
3answers
69 views

Question about the exponential function.

For $x\in\mathbb R$ we define $$\exp(x) := \sum_{n=0}^\infty \frac{x^n}{n!}. $$ This is the standard definition of the exponential function, e.g. given by Rudin in the introduction to Real and Complex ...