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
40 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
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1answer
31 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} \frac{\bigg(3+(-1)^n\bigg)^n}...
0
votes
3answers
60 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 ($R=\...
1
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2answers
57 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
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0answers
34 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 ...
6
votes
6answers
80 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?
2
votes
1answer
126 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
36 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) x^k(1-x)^{100-k}...
0
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1answer
102 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) ...
6
votes
1answer
460 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
116 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, $$\frac{a_1}{a_0}.\frac{a_2}{a_1}...\frac{a_n}...
2
votes
2answers
34 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
54 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 that $...
0
votes
1answer
78 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 $n\to \...
2
votes
4answers
77 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?
1
vote
2answers
89 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
169 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
37 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
107 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 ...
9
votes
3answers
3k 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 $y(...
2
votes
3answers
78 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 ...
0
votes
1answer
41 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 n-...
0
votes
0answers
49 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 ...
8
votes
1answer
144 views

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

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
votes
5answers
240 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
vote
4answers
168 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 ...
1
vote
1answer
54 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 $\displaystyle\sum_{i=0}^{\...
0
votes
1answer
35 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
34 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
70 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 ...
10
votes
4answers
164 views

Bernoulli Number analog using Cosine

I know that Bernoulli Numbers can be found with the generating function $$\frac{x}{e^x-1}=\sum_{n=0}^{\infty}\frac{B_n}{n!}x^n$$ I was wondering if any work has been done using a similar equation $$\...
1
vote
1answer
29 views

Exercise on inequalities in bounded derivatives (from Spivak)

Suppose $f$ is two times differentiable in $(0,\infty)$ and that: $|f(x)| \leq M_{0}, \forall x>0$; $|f''(x)| \leq M_{2}, \forall x>0$. a) Show that $$|f'(x)| \leq \frac{2}{h} ...
2
votes
1answer
333 views

Estimating accuracy of Taylor series approximations with 2 bounds

I have a question from a previous exam as such: Use Taylor's Inequality to estimate the accuracy of the approximation $f(x) \approx T_{3}(x)$ when $0.8 \leq x \leq 1.2$. I computed from an ...
4
votes
1answer
37 views

When variable substituitions are allowed in Taylor's Polynomials and when they aren't?

Let $f:[-a,a] \rightarrow \mathbb{R}$ be a function assuming derivatives up to the $n$-th order in the open interval $(-a,a)$. The Taylor polynomial of $f$ around $0$ is: $$P_{n}(x) = \sum_{k=0}^{n} \...
0
votes
2answers
82 views

Summation of a Geometric Power Series

I was told that I could simplify the power series into a a^k geometric series by setting a= (6*e^t)/(11^n). I remember from Calculus 2 that I could calculate the sum of a geometric series in this form ...
0
votes
2answers
57 views

Defining a Function using the Power Series

. I understand c & d as fairly simple first and second derivatives of a multivariable function @ t = 0. But it escapes me on whether it is acceptable to separate the functions and do the product ...
3
votes
2answers
274 views

A “generalized” exponential power series

I'm wondering if $$ e^x = \sum_{k=0}^\infty \frac{x^k}{k!} $$ what would this be $$ \sum_{k=0}^\infty \frac{x^{k+\alpha}}{\Gamma(k+\alpha)} = \large{?}_{\alpha}(x) $$ for $\alpha \in (0,1)$? ...
1
vote
1answer
30 views

proving with a sequence

The question is : Show that if $n$ is a power of $2$, then $$\sum_{i=0}^{\log_2n-1}2^i=n-1\;.$$ Tried induction at first and tried to prove it on 2n but nothing came out of it. Then i tried ...
1
vote
1answer
86 views

sum of a complex power series

I have to find the sum of a complex power series inside radius of convergence, for simplicity let's say the series looks something like that: $f(z)=\sum_{n \geq 2} \frac{z^n}{n(n-1)}$ Then after ...
0
votes
1answer
45 views

Why can you use the Maclaurin Series for certain cases of function not about 0?

Is it possible to use the Maclaurin Series in a problem like this one (AP Calculus BC Question 6 from a few years ago)? Write the first four nonzero terms and the general term of the Taylor ...
1
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2answers
121 views

Need help understanding power series

So as I understand so far: A power series is like any other series except now the partial sums depend on the variable x. The value of x determines the convergence or divergence of the series, meaning ...
2
votes
4answers
271 views

Evaluate the sum $\sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)!}$

$$\sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)!}$$ I am having difficulty finding the function that represents this series. I have only found radius of convergence which is $(-\infty,+\infty)$ from the ...
0
votes
1answer
36 views

Radius of convergence of $\sum_{n=1}^{\infty }n!(2x-1)^n$

I get to the point of $\lim_{x\rightarrow \infty }(n+1)\left | 2x-1 \right |$ using the ratio test. It looks like it should always diverge but I'm not sure. Also not sure what the x=0.5 case does.
1
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1answer
290 views

Generating Functions - Extracting Coefficients

In many counting problems, we find an appropriate generating function which allows us to extract a given coefficient as our answer. In cases where the generating function is not one that is easily ...
1
vote
1answer
57 views

Radius of convergence of a power series whose coeffecients are “discontinuous”

I have a power series: $s(x)=\sum_0^\infty a_n x^n$ with $a_n= \begin{cases} 1, & \text{if $n$ is a square number} \\ 0, & \text{otherwise} \end{cases}$ What is the radius of convergence ...
0
votes
1answer
28 views

Formula for weighed geometric sum

I'm trying to find an easy way to derive a formula for: $S_{n} = \frac{1}{n}\sum_{i=0}^{n}(n-i)x^{i}$ I've found a recurrence relationship of sorts: $S_{n+1} = \frac{xnS_{n}+n+1}{n+1} = x\frac{n}{n+...
1
vote
0answers
56 views

Pointwise and uniform convergence of series of functions

Consider power series $f(x)=\sum\limits_{n=0}^\infty c_nx^n$. Let $\alpha = \limsup\limits_{n\to\infty} \vert c_n\vert^{1/n}$. Recall radius of convergence $r=1/\alpha$. (i) Assume $r>0$. Show ...
3
votes
3answers
118 views

Is it true that $\lim \limits_{x \to a} [l(x)-m(x)] = 0 \implies l(x)$ has the same sign of $m(x)$ for $x$ sufficiently close to $a$?

This was present in a demonstration of a theorem, however, it didn't seem obvious to me. Is this a general result? Here is the exact context where this appeared (from Spivak's Calculus book): If f ...
0
votes
1answer
19 views

Infinite series with a binomial

I'd like to know, is there any place where I can find the proof of this? in some radius of convergence?
0
votes
2answers
22 views

expand function, taylors series, combinatorics, generation functions

I have to expand $f(z)$ into a formal power series $f(z) = \sum\limits_{k=0}^\infty a_kz^k$ (for $z$ close to 0) $f(z)= \frac{z^3}{1-4z+3z^2}$ I know that: $\frac{1}{1-z} = \sum\limits_{k=0}^\infty ...