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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4answers
47 views

Power series solution (Why the constant of the recurrence relation can be chosen arbitrarily?)

Please help me understand this: Solve $(x+1)y''-(2-x)y'+y =0$ First, since $x_0=0$ is an ordinary point, it can be guaranteed that we can find two independent power series solutions centered at ...
1
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1answer
23 views

Is there any known way to sum a subserie (square indices) of geometric series?

I was interested in the following sum. Although im not sure there exist any known way to sum this...it seems rather difficult. Can we sum for $0<r<1$ $$\sum_{k=0}^{\infty}r^{k^2}= ...
0
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0answers
22 views

Evaluation of r^3 from zero to N (Sigma Notation)

I have to evaluate the following expression, $\sum_{n = 0}^{9} (n^3 -1)$. I know that $\sum_{n = 1}^{9}(n^3 -1)$ is given by $\frac{1}{4}(9)^2(9+1)^2 - 9$ But, how do I do this from zero to ...
2
votes
5answers
93 views

Example 3.40 (b) in Baby Rudin: How to find $\lim_{n\to\infty} \sup \frac{1}{\sqrt[n]{n!}}$?

Here is Theorem 3.39 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: Given the power seires $\sum c_n z^n$, put $$\alpha = \lim_{n\to\infty}\sup\sqrt[n]{\vert c_n ...
0
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1answer
10 views

Show that any polynomial of odd degree 2n+1: $f(x)=\sum_{k=0}^{2n+1} a_kx^k $, $a_{2n+1}\neq0$ has at least one real root.

Show that any polynomial of odd degree 2n+1: $$f(x)=\sum_{k=0}^{2n+1} a_kx^k $$ $a_{2n+1}\neq0$ has at least one real root. I would like to prove this using IVT, how would I go about starting ...
2
votes
1answer
46 views

Let $f$ be an analytic isomorphism on the unit disc $D$, find the area of $f(D)$

Let $f$ have power series $f(z) = \sum_{n=1}^\infty a_n z^n$ in $D$, then prove that $\mathrm{area}\, f(D) = \sum_{n=1}^\infty n \,|a_n|^2$. Note: We define $\mathrm{area}\, S = \iint_S ...
0
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0answers
36 views

Holomorphy on open unit disk and continuity to the closure implies absolutely convergence of coefficients?

I am having trouble proving that the space of holomorphic functions continuous till the closure in the unit open disk coincides with the power series whose coefficients form an absolute convergence of ...
1
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0answers
35 views

Laurent series for $f(z) = \exp(z+\frac{1}{z})$ around $0$

I need to find the Laurent series of the following function around $0$ - $$f(z) = \exp(z+\frac{1}{z})$$ Now by power series expansion, I got $$f(z) = \sum_{m=0}^{\infty} \frac{z^m}{m!} ...
0
votes
2answers
40 views

For what values of $k$ to both of the following series converge?

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
0
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2answers
36 views

Uniform Convergence of $\sum_{i=1}^\infty \arctan\left(\frac{x}{i^2}\right)$ and its differentiabilty

I was trying to prove it is uniform convergent by it is Cauchy in sup-norm, since I don't know what does it converge to and it seems that M-test fail (as each term is bounded by $\pi/2$). ...
1
vote
1answer
18 views

Is this an incorrect error bound value?

In Step 3, they are determining the $(n+1)^{th}$ term. I think the proofreader just added 1, instead of subbing in (n+1). Is that right? I think the correct term should be ...
0
votes
1answer
36 views

Radius of Convergence of $\sum_{n = 0}^{\infty} \frac{(-1)^nn!x^n}{n^n}$

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
0
votes
1answer
26 views

How would I show $|x| \le 1$ given the equation for $x$ the expression in the equation?

The expression is $x = \sin(\theta /2)$. I am asking how would I show that $\sin(\theta/2)\le1$ based on the expression? I already know that the biggest $\sin$ will ever get is $[-1, 1]$ which is the ...
3
votes
3answers
126 views

How is the last “=” true?

How can the last equality be true? $$ G(u)=\frac{g}{(1+u)^g-1}-\frac1u=\frac{g}{gu + \cdots + u^g} - \frac{1}{u}=\sum_{k=0}^\infty c_ku^k $$
1
vote
1answer
14 views

What is the Laurent expansion of f(z)=1/(z-3)?

What is the Laurent expansion of f(z)=1/(z-3)? In the region, ㅣZ-3ㅣ>0 ? I just computed the Laurent expansion in the region ㅣZㅣ>3 by dividing the denominator by 1/z and making it as a geometric ...
-3
votes
0answers
26 views

4th derivative of $1 - 9x + 16x^2 - 25x^3 + \dots$ [on hold]

Fined the 4th derivative of $f$ at $x=0$ given that the MacLaurin series of $f$ is $f = 1 - 9x + 16x^2 - 25x^3 + \dots$.
1
vote
5answers
51 views

Does this power series $\sum_{n=0}^{\infty} \frac{n^n}{(n!)^2}x^n$ converge for all $x$?

Does this power series $\sum_{n=0}^{\infty} \frac{n^n}{(n!)^2}x^n$ converge for all $x$? It was told to me that the series does converge for all $x$, however I have investigated with a computer ...
-3
votes
0answers
19 views

Determining the r value for a series solution [on hold]

I have trouble answering the 2nd part of the question. For the 1st part, I just simply plugged it into the derivatives and I end up getting $r_{1}=1/2$ and $r_{2}=1/3$. For the 2nd part, I tried ...
6
votes
4answers
102 views

Power series solution for ODE

The ODE I have is $$y'(x)+e^{y(x)}+\frac{e^x-e^{-x}}{4}=0, \hspace{0.2cm} y(0)=0$$ I want to determine the first five terms (coefficients $a_0,\ldots, a_5$) of the power series solution ...
0
votes
0answers
17 views

Second Order Linear Non-Homogeneous DE solution with Power Series $x^2y'' - 4xy' + 6y = x^2 \cos x$

My instructor wants me to solve the above equation using power series and another method, and then to confirm the results are the same This equation does not have constant coefficients and a can't ...
0
votes
2answers
41 views

Solve $y'=e^{x^2}y$ (with $3$ terms only) in using power series

Solve $y'=e^{x^2}y$ (with $3$ terms only) in using power series. I know that $e^{x^2}=\sum_{n \geq0} \frac{x^{2n}}{n!}$, but I don't know how to find the coefficients $a_n$ in considering ...
0
votes
1answer
51 views

Taylor expansion of $\frac1{\|x-y\|}$

Let $0\neq y\in \mathbb{R^3}$ define a function $f$ on $\mathbb{R^3}$ as $$ f(x) = \frac1{\| x-y\|} $$ What are derivatives of $f$ in zero? Or equivalently, what is the Taylor series of $f$ at ...
1
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1answer
33 views

Frobenius Method to Solve a Differential Equation

Having the equation $$x^{2}y''+xy'+x^{2}y=0$$ I get the indicial equation at get r=0, and am left with the equation. ...
0
votes
1answer
39 views

Help with generating functions

I've got two questions. I'm trying to extract the "coefficients" of a power series. I think my terminology is incorrect here but here is what I mean. Here are some examples A(Z) = 1/(1-Z) ...
3
votes
2answers
166 views

Is it true that $log(i) = \frac\pi2i$ ? If so, are both of these legitimate proofs? They seem too beautiful not to be…

Sorry if this is a naive question. I have not yet taken any upper level math courses involving complex numbers. However, in preparation for those courses, together with utilizing the knowledge that ...
2
votes
2answers
53 views

$(x-x_0)^0$ in power series [duplicate]

When I first studied power series in high school, the teacher gave the following general definition: \begin{equation} f(x)=\sum_{n=0}^{\infty}a_n (x-x_0)^n \end{equation} He then proceeded to ...
2
votes
2answers
28 views

Difficulty finding a power series representation

I have to find a power series representation and interval of convergence for $$f(x) = \frac{x-x^2}{(1+2x)^3}$$ Noting that $\frac{1}{1+2x}=\frac{1}{1-(-2x)}=\sum_{n=0}^\infty(-2x)^n$, I start taking ...
0
votes
1answer
19 views

If $f(x)=\sum_{0}^{\infty}b_n(x-5)^n$ for all $x$, write a formula for $b_8$.

If $f(x)=\sum_{0}^{\infty}b_n(x-5)^n$ for all $x$, write a formula for $b_8$. Now I know that $b_n=\dfrac{f^{(n)}(5)}{n!}$. I have tried various things but I think there is something wrong with my ...
1
vote
1answer
34 views

How to expand the summation term with power?

How to expand the following: $$ \left( \sum^{M}_{m=0} \frac{x^{m}}{m!} \right)^{K} $$ where $M$ and $K$ are positive integers.
0
votes
2answers
42 views

Expand $f(x)=\log(x+ \sqrt{1+x^2})$ into power series and determine convergence intervals

We have $f(x)=\log(x+ \sqrt{1+x^2})$ and we need to expand it into power series, which I suppose is easy because $f'(x)=(1+x^2)^{-1/2}= \sum_{k=0} {-\frac{1}{2} \choose k}x^{2k}$. It follows that ...
1
vote
1answer
32 views

Proving that a function is real-analytic

I try to solve the following exercise: Let $f:\mathbb{R}\to\mathbb{R}$ with $f(x):=\frac{1}{1+x^4}$. Prove that $f(x)$ is real analytic and compute the radius of convergence of it's Taylor series at ...
0
votes
0answers
10 views

Power-Series strictly convexity

Watch the power-series $B(\beta):=\sum_{i=0}^{\infty}b_{j}e^{\beta\cdot j}$ with $b_{j}\geq 0$ for $0<\beta<r$ where $r$ is the radius of convergence. At least one $b_{j}$ for $j\geq 2$ is non ...
0
votes
1answer
29 views

Power series solution to $y' = y(1-y)$

Find the first five terms of the power series solution to the differential equation: $$y' = y(1-y)$$ Letting $y = a_0+a_1x+a_2x^2+a_3x^3+...$ It's evident that: $$y' = \frac{dy}{dx} = ...
0
votes
1answer
48 views

Show that $e^{\varepsilon |x|^{\varepsilon}}$ grows faster than $\sum_{k=0}^{\infty} {|x|^{2k}}/{(k!)^2}$

I am wondering whether we have for $$f(x):=\sum_{k=0}^{\infty} \frac{|x|^{2k}}{(k!)^2} $$ that $$\lim_{x \rightarrow \infty} \frac{e^{\varepsilon |x|^{\varepsilon}}}{f(x)} = \infty$$ for any ...
1
vote
2answers
48 views

Show that a power series is analytic inside its radius of convergence

Let $f(z)=\sum_{k=0}^\infty a_k(z-z_0)^k$ with radius of convergence $R$ then $f$ is analytic on the open disk around $z_0$ with radius $R$. What I was thinking about is an approach based on this ...
0
votes
1answer
27 views

Expansion of $f = \sum_{n=1}^\infty \frac{1}{2^n} \frac{z^n}{1 - z^n}$ in power series around $z = 0$

Let $f = \sum_{n=1}^\infty \frac{1}{2^n} \frac{z^n}{1 - z^n}$, for $z \in \mathbb C \setminus ${$z \in \mathbb C: \exists n \geq 1,\quad z^n = 1$}. By the ratio test, the series converges when ...
2
votes
1answer
35 views

Consider the power series $\sum_{n=1}^\infty(-1)^{n-1}\frac{x^{2n+1}}{(2n+1)(2n-1)}$.

Consider the power series $\sum_{n=1}^\infty(-1)^{n-1}\frac{x^{2n+1}}{(2n+1)(2n-1)}$. Find a closed form expression for all x which converge and hence evaluate ...
0
votes
2answers
53 views

Show that entire function $f$ is a polynomial of degree at most $n$

Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be a entire function. Suppose that there are $M$, $r>0$ and $n\in \mathbb{N}$ such that $\left|f(z)\right|<M\left|z\right|^n$ for all $z \in \mathbb{C}$ ...
1
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2answers
17 views

Why is it that the interval of convergence is half open?

I am given the following power series and asked to find the radius of convergence and determine the exact interval of convergence $$\sum\biggr(\frac{3^n}{n\cdot 4^{n}}\bigg)x^n \Leftrightarrow ...
1
vote
2answers
24 views

What is the power series and domain for this function?

$$f(x)= \frac{x}{1+5x^2}$$ I got the power series: $$\sum_{n=0}^\infty (-1)^n (5^n)(x^{2n+1})$$ Assuming this is correct I would think the domain would be $$(-5^{1/3}, 5^{1/3})$$ because the absolute ...
0
votes
0answers
20 views

Continuity of series implies continuity of coefficients?

For each $t\in [0,1]$ let $f_t(z)$ be an entire function. By holomorphicity it equals its own Taylor series: $$f_t(z) = \sum_{n=0}^\infty a_n(t)\,z^n, \qquad \forall \,z\in\mathbb{C}.$$ Now ...
0
votes
0answers
19 views

Analyticity of $\frac{1}{w-z}$

Show that for every $w\in\mathbb C$ the function $f(z)=\frac{1}{w-z}$ is analytical on $\mathbb{C}\setminus \{w\}$. Proof. Let $z_0\in\mathbb{C}\setminus\{w\}$ be arbitrary and using the ...
0
votes
0answers
28 views

Solving a power series centered at 0 by integrating another power series

Problem: Find the power series for g(x) centered at 0 by integrating the power series f(x). Give answer in sigma notation along with the first four nonzero terms. $$g(x)=\ln{(1-3x)}\space \land ...
0
votes
1answer
27 views

Solving problems involving powers

How to reach from $1+𝐸𝐴𝑅= [1+𝑇×𝐴𝑃𝑅]^1/​t $ the power is (1/T) to $$APR = \frac{\ (1+EAR)^T - 1 \ }{T}$$ $$1+EAR=[1+T\times APR]^{1/T}\\ APR=\frac {(1+EAR)^T-1}T$$ and the same goes ...
11
votes
1answer
699 views

Equation about generating functions and subfactorial

Suppose $G_n(w)$ is a formal power series (really a probability generating function, see the following explanation) of variable $w$, try to solve out $G_n(w)$ for all $n\ge0$ from the ...
0
votes
1answer
30 views

Power series expansion of a complex function

How can I depict function $f(z)=\sqrt{1+\sqrt{1+z^2}}$ as a power series around zero? Where Log is a function going from $C \setminus (-\infty,0]$? Since with such a logarythm $Re(\sqrt{z}>0$ then ...
5
votes
2answers
54 views

How to evaluate $\lim\limits_{x\to 0}\frac{e^{\arctan{(x)}}-xe^{\pi x}-1}{(\ln{(1+x)})^2}$?

How to evaluate $\lim\limits_{x\to 0}\frac{e^{\arctan{(x)}}-xe^{\pi x}-1}{(\ln{(1+x)})^2}$? So I think we expand to $x^2$ since the lowest term for $\ln(1+x)$ is $x$ Let $u=\arctan{(x)}$ ...
1
vote
0answers
39 views

Radius of power series.

Consider the formal power series in one complex variable $z$ of the form $$f(z) = \sum_{n = 0}^{\infty} c_{n} (z-a)^{n}$$ where $a,c_n\in\mathbb{C}.$ Then the radius of convergence of $f$ at ...
0
votes
1answer
10 views

How to compute taylor series of $\sin{x}$ about $a=\frac{\pi}{4}$?

How to compute taylor series of $\sin{x}$ about $a=\frac{\pi}{4}$? We know $\sin{x}=\sum^{\infty}_{n=0}\frac{(-1)^nx^{2n+1}}{(2n+1)!}$. Let $t=x-\frac{\pi}{4}$, then $t+\frac{\pi}{4}=x$ Then ...
0
votes
1answer
33 views