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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4answers
71 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
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2answers
23 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
64 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
27 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
62 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
votes
1answer
89 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
votes
3answers
69 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
28 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 ...
0
votes
0answers
25 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 ...
6
votes
1answer
59 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
votes
5answers
216 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
58 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
35 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
votes
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
vote
0answers
31 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 ...
8
votes
4answers
129 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
24 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 ...
2
votes
1answer
24 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
33 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
33 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 ...
0
votes
2answers
45 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
199 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
29 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
23 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
34 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
vote
2answers
25 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
211 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
18 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.
0
votes
1answer
39 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
0answers
33 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
24 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} = ...
1
vote
0answers
36 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 ...
0
votes
1answer
17 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
19 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 ...
0
votes
1answer
26 views

Let $y,z$ be two functions such that $y=zf(y)$ where $f$ is a power series wrt $y$. Then we can find of powers series for any $g(y)$

Let $y,z$ be two functions such that $y=zf(y)$ where $f$ is a power series with respect to $y$. Then we can write any $g(y)$ as a power series of $z$ such that $g(y)=\sum_{k=1}^{\infty} ...
3
votes
0answers
25 views

Does having a real valued cauchy sequence on a function in a compact space imply the function is continous on that space?

I had to prove for a homework assignment this function $$ s_n(x) = \sum_{i=0}^n (-1)^i \frac{ x^{2i+1}}{(2i+1)!} $$ is a Cauchy sequence with respect to the sup norm for $$ s_n : [-M,M] ...
1
vote
2answers
33 views

Laurent series expansion for $\lvert z\rvert >1$.

I have a simple complex function like this: $$\frac{z+1}{z-1}$$ When I expand it by its Maclaurin series: $$\frac{z+1}{z-1} = \frac{z-1+2}{z-1} = 1 - \frac{2}{1-z} = 1 - 2\sum_{k=0}^{\infty}z^{k} ...
2
votes
2answers
82 views

Solution to ODE using Power Series

I'm currently trying to wrap my head around how to solve an ODE with series. The problem I am working on is this: Find the indicated coefficients of the power series solution about x=0 of the ...
1
vote
3answers
56 views

Problem with a simple convergent power series

Create a power series around $x=0$ and find its radius. What is $f(1)$? $$f(x)=\frac{1}{2+x}$$ Well according to me it is easy to see that the radius is $\left|x\right|<2$ and ...
3
votes
1answer
45 views

Radius of convergence $\sum\limits_{n\ge0}q^{n^2}z^n$

Radius of convergence of the power series $\sum\limits_{n\ge0}q^{n^2}z^n$ By a theorem the radius is $\limsup \lvert q^{n^2}\rvert^{1/n}=\limsup\lvert q^n\rvert=\begin{cases}0,&\text{if}\ ...
1
vote
2answers
36 views

Find a closed form for $\Sigma_{k = 1}^\infty 3x^{3k -1}$

Find a closed form form and the largest set on which this formula is valid. a) $\Sigma_{k = 1}^\infty 3x^{3k -1}$ Attempt: $\Sigma_{k = 1}^\infty 3x^{3k -1} = 3\Sigma_{k = 1}^\infty x^{3k} x^{-1} = ...
2
votes
2answers
52 views

Finding power series solution to differential equation $(1-x)y'=y$ centered at $ x=0$

I'm trying to teach myself how to solve differential equations with power series. I am stuck on working through $$(1-x)y'=y \text{ centered at } x= 0.$$ I've gotten to the point where I must figure ...
1
vote
1answer
59 views

Radius of convergence of power series (random variables)

Let $z_n$ be a sequence of independent identically distributed random variables and let $$f(x)=\sum_{n=0}^\infty z_nx^n$$ be a random power series. How to show that the radius of convergence is ...
2
votes
2answers
49 views

Integrating a power series proof

So I began with using the hint and got to $f(x) - \sum_{n=?}^{\infty} a_nx^n$ But I have a question about the ? in the sum obviously. Because it's been differentiated, do we have to increase the ...
0
votes
1answer
19 views

Determining power series convergence radius

I try to determine the convergence of the following power series $$ \sum_{k=1}^\infty \frac{x^k}{\sqrt{k(k+1)}}$$ I tried it using the ratio test, but I am not sure if this is correct My approach: ...
0
votes
1answer
39 views

Power Series with singularities in {z: |z|=1}

Prove that all the points in $D=\left\{ z \in \mathbb{C} : \mid z\mid=1 \right\}$ are singularities of the function $$ f(z)=\sum_{n=0}^{\infty} \frac{z^{n!}}{n!} $$ This was easy for the ...
0
votes
0answers
11 views

I need help finding the radius of convergence along with the interval of convergence

The power series in question is: I used the root test to find the interval of convergence and got [-6/5,-4/5) however I'm not sure how to find the radius of convergence.
0
votes
2answers
16 views

I need help finding the interval of convergence for a power series

I need to find the interval of convergence for the following power series: I used the ratio test and got (2X)lim(n/n+1) and this is pretty much where I'm stuck.
2
votes
1answer
31 views

radius of convergence of a complex power series

Can you tell me what you think about my solution to this problem? In case it's wrong, or needs changes, just something like "try looking at this", "consider that"... a hint is enough, please no ...