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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2answers
37 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} ...
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0answers
7 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) = ...
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1answer
48 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 ...
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0answers
21 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 ...
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0answers
11 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 (...
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1answer
36 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 ...
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2answers
74 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 $\lambda$ ...
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0answers
40 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} = ...
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1answer
29 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 ...
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1answer
24 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 ...
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1answer
37 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.!
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0answers
32 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 $$\sum_{k=1}^\infty \frac{1}{q^{k(k+n)}}, \qquad\qquad \sum_{k=1}^\infty \frac{k}{q^{k(k+n)}}, \qquad\qquad \text{ for }q>1,\;n\in \mathbb{N}?$$ ...
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3answers
34 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 ...
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2answers
26 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 ...
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3answers
37 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} ...
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1answer
43 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 ...
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3answers
34 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}= ...
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0answers
18 views

How to find a sum for power series in a given interval [closed]

for this power series, first thing I tried to do was try to take integrals term by term and then find the definite integral in the given interval this is the integral that I find, but here is my ...
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2answers
16 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 ...
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2answers
19 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 ...
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1answer
22 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} ...
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3answers
53 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 ...
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0answers
16 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 - ...
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3answers
42 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 ...
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0answers
14 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
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4answers
53 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?
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1answer
20 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 ...
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1answer
32 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) ...
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1answer
14 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) ...
4
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1answer
53 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 ...
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3answers
64 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
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2answers
23 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
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2answers
47 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 ...
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1answer
35 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
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4answers
70 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?
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2answers
19 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 ...
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1answer
63 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 ...
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0answers
25 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
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3answers
56 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 ...
4
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1answer
85 views

Solution of $y''+xy=0$

The differential equation $y''+xy=0$ is given. Find the solution of the differential equation, using the power series method. That's what I have tried: We are looking for a solution of the form ...
2
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3answers
67 views

How do you solve $10^{-3/20}$ as $1/2^{1/2}$?

My electronics lecturer was able to instantly solve $10^{-3/20}$ as $1/2^{1/2}$, but he was not able to explain it to me because he said that it was just a number he was very familiar with. FYI, the ...
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1answer
25 views

Using a power expansion to find the n-th power of a matrix

Given that $\sum_{n}^{\infty} \mu^{n} M^{n} = (I-\mu M)^{-1}$ Wherein $\mu$ is a scalar, $M$ is a matrix and $I$ is the identity matrix of the same dimension as $M$. How do I use this to find the ...
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0answers
24 views

why does this power series converges to sinh(x)?

given the infinite sum $$\sum_{n=0}^\infty \frac{ x^{2n+1}}{(2n+1)!}$$ of course, by ratio test, it converges for reals. I know that the answer is $\sinh(x)$ and I've seen how this is derived from its ...
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1answer
53 views

Proving that $~\lim_{x\to1^-}~\bigg(\sqrt[a]{1-x}\cdot\sum_{n=0}^\infty~x^{n^a}\bigg)~=~\Gamma\bigg(1+\frac1a\bigg)$

How could we prove that $$\lim_{x\to1^-}~\bigg(\sqrt[a]{1-x}\cdot\sum_{n=0}^\infty~x^{n^a}\bigg)~=~\Gamma\bigg(1+\frac1a\bigg)$$ for $a>0$ ? The inspiration came to me while trying find a ...
4
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5answers
213 views

Calculation of limit without stirling approximation

$\lim n^n/(e^nn!)=0$ using Stirling approximation it is obvious. But can we do it without using Stirling approximation. Now series with terms $x^n n^n/n!$ has ROC $1/e$. What we can say about ...
1
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4answers
53 views

evaluate convergent power series

given $$\sum_{n=0}^\infty x^{n} (n^{2} + n)$$ so using ratio test I have proven that it converges if and only if $$|x| < 1$$ but I'm not sure how to evaluate this infinite sum. so I thought ...
0
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1answer
32 views

Series representation of hypergeometric function reciprocal?

Basically, can you represent $\dfrac{1}{_2F_1(a,b;c;z)}$ as some kind of power series? EDIT: This question came from something I was doing with generating functions were ...
0
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1answer
29 views

$h(x) = \sum_{n=0}^\infty c_n x^n$ show h(x)=f(x)g(x)

Given $f(x)= \sum_{j=0}^\infty a_j x^j$ and $g(x)= \sum_{k=0}^\infty b_k x^k$. $f(x)$ has radius of convergence $R_1 > 0$ and $g(x)$ has radius of convergence $R_2 > 0$ Let $c_n= \sum_{j=0}^n ...
1
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0answers
28 views

Bernoulli Number analog using Cosine (part 2)

Earlier today I posted this inquiry about the function below: $$\frac{x^2}{\cos{x}-1}=\sum_{n=0}^{\infty}\frac{C_n}{n!}x^{2n}$$ I got some good feedback but as I was playing around, I wondered if ...
2
votes
2answers
49 views

Taylor series for the function $f(z) = \frac{1}{(z-5)(z-7)}$ on a disc centered at point $z_0=3$

I started by expressing the function as sum of two fractions using partial fraction decomposition to get $\frac{-1}{2(z-5)} + \frac{1}{2(z-7)}$ However I could only then end up writing that as the ...