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).

learn more… | top users | synonyms

-1
votes
3answers
42 views

Series solution to a 2nd order linear ODE

Consider the ODE $$(1+x^2)y''+3xy'+y=0$$ Find a solution in the form of $y=\sum a_nx^n$. So after doing the algebra using these 2: $$y'=\sum_{n=1}^\infty n a_n(x-x_0)^{n-1}$$ $$y''=\sum_{n=2}^\infty ...
1
vote
1answer
19 views

Compound interest problem with increasing deposits

An Investor starts with an initial investment : $A$ He earns a steady profit of 10 percent per year. But every year he adds additional amount which increases by 15 percent every year. At the end of ...
0
votes
0answers
19 views

Solving Power Series Equality

I'm trying to solve the following power series equation to get the coefficients $\{a_k\}$: $$\sum_{k=1} ^ \infty a_k x^k = \sum_{\ell = 0} ^\infty \frac{x}{\ell !} \left( \sum_{k=1} ^ \infty a_k x^k ...
0
votes
2answers
49 views

Showing that a generating function is equivalent to some fraction

I am working with generating functions and am required to prove that the generating function for the sequence $\{a_k\}$ where $a_k = (-8)^k$ for all integers $k\geq0$ is $\cfrac{1}{1+8x}$ and I have ...
0
votes
0answers
17 views

What series describes the feedback of a fully connected network with signal strength at each node converging to 1?

Take a fully connected network with $N$ nodes operating in lockstep. One and only one node will receive a signal from an external source at a time step $t$, but the node receiving it is random. The ...
3
votes
0answers
25 views

Intuitive explanation of proof of Abel's limit theorem

Assume the series $$f(x)=\sum_{n=0}^{\infty}a_n x^n$$ converges for $-r<x<r$. Abel's theorem says that if the series also converges at $x=r$ then $\lim_{x\to r-} f(x)$ exists and we have ...
0
votes
2answers
50 views

Abel's theorem - examples

I have got Abel's Theorem in this form: If a power series is converges at one of the ends of the partition of convergence, its sum is continuous at this point (one-sided). And I have got an example ...
0
votes
0answers
9 views

Power series uniqueness

I'm having trouble with a problem for my thesis. I have two analytical functions which to first order are different, does it mean that they are different functions? Remember they have a series ...
1
vote
0answers
34 views

A generalization of Clausen's formula

Clausen's formula, $${}_{2}F_{1}(a, b; c; x)^2 = {}_{3}F_{2}(2a, 2b, a + b; 2a + 2b, c; x), \quad c = a + b + \frac{1}{2},$$ is well known. Does anyone know if this formula has been generalized for an ...
1
vote
1answer
48 views

Determining a power series of $f(z)=\exp(z^{2})$

Let $f(z)=\exp(z^{2})$. Determine a power series of $f$, i.e the coefficients $a_{k}$ such that $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ for all $z\in \mathbb{C}$, where $a_{k}=f^{(k)}(0)/k!$. First ...
0
votes
1answer
60 views

How does one obtain the expansion of $e^{-x^2}$ in a power series?

So I know that the Power Series $y = \displaystyle\sum_{m=0}^\infty\displaystyle\frac{(-1)^m}{m!} x^{2m}$ is equivalent to $e^{-x^2}$. Could someone show me why this is?
0
votes
1answer
35 views

Power series and Fourier identity approximated in two or three iterations

I understand that Fourier has proven that the sum of sines and cosines can be used to describe (almost) any curve. The power series describe that the sum of polynoms can be used to describe (almost) ...
2
votes
3answers
55 views

What's the radius of convergence for power series of $1/(1-(x+x^2))$? Is it symmetrical?

When reading the section on composition of power series in Book Calculus With Analytic Geometry (George F. Simmons) (2nd edition, pp. 517), the author claimed you can replace $x$ in a power series ...
6
votes
1answer
67 views

Can we characterize the space of functions which is real analytic but not real entire?

A complex valued function $F,$ defined on an open set $E$ in the plane $\mathbb R^{2}$, is said to be real-analytic in $E$ if to every point $(s_{0}, t_{0})$ in there corresponds an expansion with ...
1
vote
1answer
35 views

Series Solutions Near an Ordinary Point

I am attempting to solve this problem for practice: $y"-(x-3)y' - y = 0$ at $x_{0} = 3$. But it appears as though I don't have an idea of the best approach to employ to go about solving it. Can ...
1
vote
1answer
38 views

Can I solve an Euler differential equation by using the Frobenius method?

I'm having some trouble by trying to solve Euler equations by using the Frobenius method. For example, I'm asked to solve the Euler differential equation $$ x^2y'' + xy' - y = 0 $$ using a power ...
0
votes
0answers
12 views

Series Solution of Linear Second Equations - Difficulty Formatting Final Answer

I have been working the following differential equation: \begin{align} (1-x^2)y'' -8xy' -12 y = 0 \end{align} which has solution \begin{align} y= a_{0}\sum_{m=0}^{\infty}(m+1)(2m+1)x^{2m} + ...
1
vote
1answer
28 views

Index of Summation Shift? Power Series and Differential Equations

I have never had to index shift a summation series before, and it seems relatively straightforward, however, I am looking at an example in my textbook that doesn't make sense. I am wondering if ...
0
votes
0answers
27 views

Power series with complex variables inequality

I am struggling to prove the following inequality: For $z \in \mathbb{C}, r \in \mathbb{R}, n \in \mathbb{N}$, if $|z| \leq r$ and $1 \leq r < n$ then ...
6
votes
3answers
137 views

Use Taylor Series method to solve $y''-2xy+y=0$

I am doing some practice problems for solving second order ODEs, and I am a bit stuck on this one. Here is what I have: $y''-2xy'+y=0$ Let $y = \sum_{n=0}^{\infty} C_nx^n \implies y' = ...
2
votes
1answer
67 views

Showing that if the $n$th derivative of a function is bounded then it is real analytic

I reproduce from my lecture notes: Suppose $f$ is $C^\infty$ on $[a,b]$ with $$\left|f^{(n)}(x)\right|\leqslant M~~\text{for all}~~x\in(a,b).$$ Then $f$ is real analytic in $[a,b]$. Proof. ...
0
votes
1answer
18 views

Showing that two infinite series converge to the same value

I was preparing for my exam and came across this problem. Show that The series on the left hand side is the power series of $\ln(1+x)$ evaluated at $x=1$. This is what i've done so far. From ...
1
vote
2answers
106 views

Proof that $\dfrac{1}{e^x}=e^{-x}$ without converting it to $e^{x}e^{-x}=1$.

I want to show that $\dfrac{1}{e^x} = e^{-x}$ from the Taylor expansion of $e^x$. To express $\dfrac{1}{e^x}$ as a power series, I let: $$ \left(\dfrac{1}{0!}x^0 + \dfrac{1}{1!}x^1 + ...
1
vote
1answer
62 views

Showing that $\exp(\sum_{n=1}^\infty a_nX^n)=\prod_{n=1}^\infty\exp(a_nX^n)$ for formal power series

I've just come across formal power series and am not very fluent with them yet. I'd like to show that $\exp(\sum_{n=1}^\infty a_nX^n)=\prod_{n=1}^\infty\exp(a_nX^n)$. Can anybody help?
2
votes
1answer
83 views

Summation of exponential series [duplicate]

Evaluate the limit: $$ \lim_{n \to \infty}e^{-n}\sum_{k = 0}^n \frac{n^k}{k!} $$ It is not as easy as it seems and the answer is definitely not 1. Please help in solving it.
1
vote
1answer
38 views

Radius of convergence of power series $\sum_{n=1}^{\infty}{\frac{\sin n!}{n!}} {x^n}$

The power series $\displaystyle\sum_{n=1}^{\infty}{\frac{\sin n!}{n!}} {x^n}$ has radius of convergence $R$, then $R\geq1$ $R\geq e$ $R\geq2e$ All are correct I wanted to know how will get the ...
13
votes
2answers
358 views

Ramanujan's approximation for $\pi$

In 1910, Srinivasa Ramanujan found several rapidly converging infinite series of $\pi$, such as $$ \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}. ...
3
votes
1answer
61 views

Showing $y_1$ or $y_2$ are not polynomials

proof that $y_1$ or $y_2$ are not a polynomial for any $n$ $$ y_1(x)=1-\frac{n(n+1)}{2!}x^2+\frac{(n-2)n(n+1)(n+3)}{4!}x^4-+\cdots$$ $$ ...
4
votes
1answer
83 views

Short form of few series

Is there a short form for summation of following series? $$\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^kn!\alpha^{2n}((2y-1)^{2k+1}+1)}{2^{2n+1}(2n)!k!(n-k)!(2k+1)}$$ ...
1
vote
1answer
86 views

Product of two $2$-variables Taylor series

Using the standard multi-index notation, suppose we have the two Taylor series $$ f(\theta) := \sum_{|\alpha|=0}^{\infty} a_{\alpha} \theta^{\alpha} $$ and $$ g(\theta) := \sum_{|\alpha|=0}^{\infty} ...
3
votes
0answers
171 views

Proof on why $0-1+2-3+4-\ldots\neq-1/4$

When reviewing my notes on series' convergence, I thought of applying a workaround on why $\sum_{n=0}^{\infty}(-1)^nn$ should or shouldn't be $-1/4$ (I recalled this page). I started by considering ...
1
vote
1answer
35 views

Radius of convergence of powerseries containing $(\log n)^n$

$$ \begin{align} \sum_{n=2}^\infty (\log n)^n(z+1)^{n^2} \end{align} $$ What is the radius of convergence of this power-series? I tried applying the root test and the ratio test , but I couldn't ...
0
votes
2answers
42 views

Power series solution to integral equation

Hi guys i'm reading a paper in which the authors have two coupled integral equation for the function $f(x)$ and $g(x)$, in order to solve this problem they employ a power series expansion of these ...
0
votes
2answers
84 views

Disk of convergence of the series $ \sum\limits_{n=1}^\infty n!\,(z-i)^{n!} $

$$ \sum_{n=1}^\infty n!(z-i)^{n!} $$ Find the disk of convergence of this powerseries. Can I set $n!=k$ and then deal with $\sum_{n=1}^\infty k z^k$ . On another note $\frac{z^{(n+1)!}}{z^{n!}}$ ...
1
vote
2answers
98 views

How to obtain probability distribution from the generating function $G(s) = e^{a(s-1)^2}$?

I was trying to get the probability distribution $p(n)$ from a generating function $G(s)$ like this: $G(s) = e^{a(s-1)^2}=\sum s^np(n)$ I need first to do Maclaurin expansion of the exponential and ...
1
vote
0answers
45 views

Power series and Taylor series

Let $f:\Bbb R\to\Bbb R$ be a $\cal C^\infty$ function. Consider the power series $$\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n$$ and call $R$ its radius of convergence. Then, is it true that in ...
0
votes
1answer
26 views

Calculating $\sum_{n=0}^\infty (r e^{2 \pi i \alpha})^{n!}$ for $\alpha \in \mathbb Q$.

I need to calculate $\sum_{n=0}^\infty (r e^{2 \pi i \alpha})^{n!}$ for $\alpha \in \mathbb Q$ and $r \in \mathbb R$. My Attempt: $\sum_{n=0}^\infty (r e^{2 \pi i \alpha})^{n!}=\sum_{n=0}^\infty r ...
0
votes
2answers
29 views

Find the Taylor series and prove it converges using the defintion

I'm studying for the FE Exam. A simple walk-through would be appreciated to help my understanding of how to solve similar problems. Find the Taylor series about $x=2$ for the function $f(x) = x^5 - ...
0
votes
2answers
45 views

Radius of convergence of powerseries $\sum_{n=1}^\infty \frac{(-1^n)}{n!}z^n$

$$ \begin{align} \sum_{n=1}^\infty \frac{(-1)^n}{n!}z^n \end{align} $$ Find the radius of convergence of this powerseries. To determine the radius of convergence should I split it into two separate ...
0
votes
1answer
45 views

Power series function expansion as solution for integral equation

I'm facing an integral equation whose unknown is a function $f(x)$: The equation is of the kind: $$ K = \int_{-l}^{l} G(x,s)f(s)ds $$ So it's a Fredholm integral equation that is rewritten in this ...
0
votes
2answers
42 views

Interval of convergence using ratio test on the series ln(1 - x)

I have to find the series expansion and interval of convergence for the function ln(1 - x). For the expansion, I have gone through the process and obtained the series: -x - (x^2/2) - (x^3/3) - . . . ...
1
vote
1answer
23 views

Maclaurin series - Approximation and interval of convergence

This is a problem which I should apparently be solving with Maclaurin series, but I failed to do so. So I attempted it with binomial series, with 5 terms and an error less than the requirement in ...
0
votes
1answer
20 views

Power series function - convergence interval

Could someone help me finding the function and convergence interval for following power series? I don't need a step by step answer, but I'm not entirely sure where to start. $\sum_{n=0}^{+\infty} ...
0
votes
1answer
8 views

Sequence Interval of convergence

I could someone help me with the following sequence of functions of which I attempted to find the interval of convergence, but I couldn't get it to match with the solution I get from WolframAlpha ...
1
vote
0answers
90 views

A function equal to its Taylor series on an interval is also equal to its Taylor series on a subinterval with different center

Suppse the power series $ \sum_{n=0}^\infty a_n (x-a )^n$ has positive radius of convergence $R$ and thus defines a real analytic fuction $f$ on $(a-R,a+R).$If $x_0$ is a point with $|x_0-a|<R,$ ...
0
votes
1answer
51 views

A nonregular local ring [duplicate]

Consider the ring of the formal power series $k[[T_1,\ldots,T_n]]$ ($k$ algebraically closed) where $\mathfrak m$ is the maximal ideal. If $f\in\mathfrak m^2$, why $$\frac{k[[T_1,\ldots,T_n]]}{(f)}$$ ...
0
votes
2answers
36 views

Power Series constant values

I know that we could represent the function $\frac{8x}{7+x}$ as a power series $8\sum\limits_{n=0}^{\infty}(-1)^n(\frac{x}{7})^{n+1}$ Therefore the first few terms would be: ...
2
votes
0answers
50 views

$f\in C^\omega ((a-R,a+R),\mathbb{R})$ [closed]

We discuss the following question in the field of real numbers . A a power series $f(x)= \sum_{n=0}^\infty a_n (x-a )^n$ converges in $(a-R,a+R).$ Prove: $$\forall x_0\in\left(a-R,a+R \right), ...
0
votes
1answer
48 views

$\lim_{x\rightarrow {0}^{+}}\sum_{n=1}^{\infty}{(-1)}^{n-1}\frac{1}{n!{x}^{n}}=?$

$$\lim_{x\rightarrow {0}^{+}}\sum_{n=1}^{\infty}{(-1)}^{n-1}\frac{1}{n!{x}^{n}}=?$$ I found this question during my study. In my opinion ,it is not difficult to solve ,but it is interesting. So I ...
4
votes
1answer
92 views

'Deriving' the Laplace Transform from the $z$ Transform: Missing a $\Delta t$

Textbooks normally give the following 'derivation' (or justification, if you prefer) of the z-Transform from the Laplace Transform. Let $x(t)$ be a signal defined on $t\geq 0$, and write a discretized ...