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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3answers
45 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 ...
1
vote
1answer
23 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
29 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
11 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
18 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
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0answers
25 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
127 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
44 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
13 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
102 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
61 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
73 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
35 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 ...
1
vote
0answers
59 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
54 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
78 views

Short form of few series

Is there a short form for summation of following series? $$\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^kn!\alpha^{2n}((2y-1)^{2k+1}+1)}{2^{2n+1}(2n)!k!(n-k)!(2k+1)}$$ ...
0
votes
1answer
72 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
166 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
23 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
35 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
0answers
49 views

What is the radius of convergence for the power series of the Riemann zeta function at $x_0=0$?

Someone said about complex analysis that power series behave the way you expect in real analysis. After suggested edits in the comments below: What is the radius of convergence for the power series ...
0
votes
2answers
39 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
88 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
41 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
27 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
37 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
28 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
34 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
21 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
19 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
7 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
86 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
50 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
35 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
49 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
80 views
+100

'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 ...
3
votes
2answers
67 views

Radius of convergence of power series which has factorial term

I am trying to find radius of convergence of the following power series: $\sum_{n\geq 1} n^n z^{n!}$ I tried ratio test but it became complicated, I have never seen such radius of convergence problem ...
0
votes
1answer
34 views

Upper bound for modulus of a function

Let $f(t,x)$ be a bounded and continuous function on $\mathbb{R}_t \times \mathcal{U}$ where $\mathcal{U}$ is an open neighborhood of $0 \in \mathbb{C}_x$. Moreover, assume that for each fixed $t$, ...
2
votes
2answers
59 views

Find the first 5 coefficients of the series $\frac{6x}{x+9} = \sum_{n=0}^\infty C_n x^n$

I rewrote the equation series as $$ \frac69 \sum_{n=0}^\infty \left(\frac{-1}{9}\right)^n x^{n+1} $$ And therefore have coefficients of $C_0 = 6/9, C_1 = \left( 6/9 \right) ...
2
votes
1answer
58 views

Recurrence - using power series

Could you help me in solving this recursion( a closed form ) using power series $\mu(n)=\mu(n−1)k_0+(n−1)\mu(n−2) k_1 \tag 1$, where $k_0,k_1$ are constants $\mu(0)=3,\mu(1)=5$ HINT: We can think ...
2
votes
1answer
38 views

Showing the following function is entire…

The full problem asks about the following function using it's Maclaurin series: $$f(x)=\left\{ \begin{array}{lr} \frac{\sin(z)}{z} & : z \neq 0\\ \;\;\;\;1 & : z=0 \end{array} \right.$$ I've ...
1
vote
2answers
64 views

Summation of infinite series

If we know the series sum given below converges to a value $C$(constant) $$\sum_{n=0}^{\infty}a_n =C \tag 2$$ Can we generate following in terms of C. values of $a_n$ will tend to zero as n goes to ...
1
vote
2answers
33 views

Find first five terms of the power series representation for the function

f(x) = ${e^x cos(x^2)}$ So I have the answer which is ${1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+(\frac{1}{4!}-\frac{1}{2!})x^4+...}$ So I know that ${e^x = \sum\frac{x^n}{n!}}$ and that ...
0
votes
3answers
27 views

Finding the sum of a sequence of terms

$$1/1(2) - 1/3(2^3) + 1/5(2^5) - 1/7(2^7)$$ This is equal to $$\sum_{n=0}^\infty(1/2)^{2n+1}(-1)^n/(2n+1)$$ Differentiating this leads to: $$\sum_{n=0}^\infty(-1/4)^n$$ Which is equal to $4/5$ Thus, ...
1
vote
3answers
89 views

Does $x^2+ x^4/(3\cdot4) + x^6/(3\cdot4\cdot5\cdot6) + \cdots$ have any compact form?

Is there any compact form for the following series $$F_1(x) = x^2+ \frac{x^4}{3\cdot4} + \frac{x^6}{3\cdot4\cdot5\cdot6} + \cdots$$ $$F_2(x) = x+ \frac{x^3}{2\cdot3} + \frac{x^5}{2\cdot3\cdot4\cdot5} ...
2
votes
2answers
56 views

Evaluate Definite Integral to desired accuracy

Evaluate $$\int_0^{1/2}x^3\arctan(x)\,dx$$ My work so far: $x^3\arctan(x) = \sum_{n=0}^\infty(-1)^n \dfrac{x^{2n+4}}{2n+1}$ $$\int_0^{1/2}x^3\arctan(x)\,dx = \sum_{n=0}^\infty ...
2
votes
1answer
99 views

What's $\sum{\frac{x^n}{n^3}}$?

What's $\displaystyle f(x)=\sum_{n=1}^\infty{\frac{x^n}{n^3}}$? Note its derivative: $$\displaystyle f'(x)=\sum_{n=1}^\infty{\frac{x^{n-1}}{n^2}}$$ and the next derivative: $$\displaystyle ...
9
votes
4answers
269 views

Asymptotic behavior of $\sum\limits_{k=1}^n \frac{2^k}{k}$

I'm looking for an asymptotic equivalent of $$\sum_{0 < k \le n} \frac{2^k}{k}$$ as $n \to \infty$. A plausible candidate seems to be $\frac{2^{n+1}}{n+1}$ (WolframAlpha plot, and the intuitive ...