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
votes
1answer
48 views

Finding the function of the power series $\sum\limits _{n=1}^{\infty}\frac{x^{2n+1}}{n}$

Find the sum function of the following power series $\sum\limits _{n=1}^{\infty}\frac{x^{2n+1}}{n}$ I actually felt like I had the right idea on this one, my solution is: Moving one $x$ out of ...
0
votes
2answers
102 views

showing that the partial sums of $ \log(j) = n\log(n) - n + \text{O}(\log(n))$

I'm trying to show that the partial sums of $\log(j) = n\log(n) - n + \text{O}(\log(n))$ I know that $$\int_1^n\log(x)dx = n\log(n) - n + 1$$ so that this number is pretty close to what I want. Now ...
0
votes
2answers
70 views

Power series expansion of $x\ln(\sqrt{4+x^2}-x)$

Find $a_n $ where $x \ln(\sqrt{4+x^2}-x) =\sum_{n=0}^{\infty} a_nx^n$. I know that I must find power series expansion of $ln(\sqrt{4+x^2}$ but it doesn't help. Can anyone give me a hint? many ...
0
votes
0answers
17 views

General solution using power series method

I'm trying to get the general solution to the differential equation, $${\partial^2y(x) \over \partial x^2}+4x{\partial y(x) \over \partial x} + y(x) = 0$$ using the power series method, about the ...
0
votes
1answer
27 views

Construct Maclaurin series for $f(x)=x\sin(2x)$ in sigma notation and use this to find $f^{(14)} (0)$ and $f^{(9)} (0)$

So I used the known power series of $\sin(x)$ to get down to the Maclaurin in sigma notation. $$\sum_{n=0}^{\infty }\frac{(-1)^{n}(2)^{2n+1}}{(2n+1)!}x^{2n+2}$$ I'm a bit foggy on the $f^{(14)} (0)$ ...
6
votes
1answer
146 views

On $e^{5x}+e^{4x}+e^{3x}+e^{2x}+e^{x}+1$

Define the following, $$F_2(x) := \frac{1}{2}+\frac{(2x)}{1!} B_2\Big(\tfrac{1}{2}\Big)+\frac{(2x)^2}{2!}B_3\Big(\tfrac{1}{2}\Big)+\frac{(2x)^3}{3!}B_4\Big(\tfrac{1}{2}\Big)+\dots $$ ...
1
vote
0answers
32 views

Radius of convergence for modified series

I have the power seires $$\sum_{k=0}^\infty{a_kz^k}$$ with $z, a_k \in \mathbb{C}$. Say this series has radius of convergence $R$. I want to show that then $$\sum_{k=1}^\infty{a_k k z^{k-1}}$$ has the ...
2
votes
3answers
103 views

How to show $\sum_{k=n}^\infty{\frac{1}{k!}} \leq \frac{2}{n!}$

$$\sum_{k=n}^\infty{\frac{1}{k!}} \leq \frac{2}{n!}$$ Can someone show why this estimate holds true? I tried quite a bit but couldn't really find a way to approach this. WolframAlpha says it is true ...
1
vote
2answers
84 views

Convergence of $\sum_{n = 1}^\infty 1/n^2$.

I know that $\sum_{n=1}^\infty 1/n$ diverges whereas $\sum_{n=1}^\infty 1/n^2$ converges. Intuitively, I do not see the difference. If $n \to \infty$, the denominators in both fractions will be so ...
-1
votes
1answer
40 views

Power Series Solution of y''+2xy=0

So I've been doing some Power Series Method problems for non constant differentials but I can't seem to get this one. $$y'' + 2xy=0, \ y(0)=1 , \ y'(0)=1$$
2
votes
2answers
78 views

Branch cut for $\sqrt{1-z^{2}}$ and Taylor's expansion!

I'm working in a problem that involves the equation $$ w(z)=\sqrt{1-z^{2}} \,\, . $$ I already know that there're two branch points in this equation, namely $\pm 1$, so there's a Riemann surface ...
10
votes
2answers
275 views

Proof of $\sum_{n=1}^{\infty} \frac{x^n \log(n!)}{n!} \sim x \log(x) e^x$ as $x \to \infty$

Prove that $$\sum_{n=1}^{\infty} \frac{x^n \log(n!)}{n!} \sim x \log(x) e^x \,\,\,\text{as}\,\,\, x \to \infty$$ and $$\sum_{n=1}^{\infty} \frac{(-x)^n \log(n!)}{n!} \to 0 \,\,\,\text{as}\,\,\, x \to ...
1
vote
1answer
49 views

Proof of $\lim_{x\to\infty}\frac{\text{Ei(x)}}{e^x}=0$

I encountered the following limit while doing calculation $$\lim_{x\to\infty}\frac{\text{Ei(x)}}{e^x}=0$$ which is equivalent to $$\lim_{x \to \infty }e^{-x}\sum_{n=1}^{\infty}\frac{x^n}{n·n!}=0$$ and ...
3
votes
1answer
64 views

How can I justify that the partial sums of $\frac{(2n)^k}{k!}$ is less than the number $\frac{(2n)^n}{n!}$?

I am currently using Rouche's Theorem from complex analysis but am working on an upper bound and want to show $$\sum_{k=0}^{n-1}\frac{(2n)^k}{k!}< \frac{(2n)^n}{n!}$$ Any suggestions are ...
0
votes
1answer
47 views

Is my idea of decomposing a meromorphic function into a sum of Laurent series correct?

We know that complex-analytic functions $f(z)$ agree with their power series representations on their domain of analyticity. If a meromorphic function has simple poles at $z_1, ..., z_m$ and ...
1
vote
2answers
25 views

Power series of Isometries

I was asked to show, given an isometry U from a finite dimensional inner product space V to itself that $$ A_n(x)=\frac{1}{n}\sum_{0}^{n}U^{n}(x) \to 0 \text{ as } n\to \infty \text{ for} x\in Im(I-U) ...
0
votes
1answer
34 views

Power series expansion with coefficient explanation?

So I'm doing power series expansion and I have this example in my notes to walk through the steps of solving a differential equation using power series. So I was just wondering if someone could ...
0
votes
0answers
15 views

Expanding two products term by term

I have the following double product $$ \prod_{a=0}^{3}\prod_{b=0}^{3-i} \Big((p-b)u + (q-a)v\Big) $$ and it does not matter what these variable really are. I want to analytically expand it so I first ...
0
votes
1answer
23 views

Simple power series convergence question

So for my particular power series, I find that my interval of convergence is $-3 < x < 3$, so $R = 3$. I do a nth term test on the original equation with -3 and 3 and find that the series ...
3
votes
1answer
37 views

Root of an equation given by an infinite series

Consider the equation $$e^{-x} = x-1 $$. We know that there is only one real root, $r$. How can it be shown that $$r= 1 + \sum_{n=1}^{\infty}\frac{(-n)^{n-1}e^{-n}}{n!}$$
0
votes
0answers
22 views

Product with multi-dimensional matrix

We know the power series in scalar case $f:\mathbb R\to\mathbb R$ $$ f(x)=a_0x^0+a_1x^1+a_2x^2+a_3x^3+\dots $$ so what is the extension to the multi-dimensional case $f:\mathbb R^n\to\mathbb R$ $$ ...
1
vote
4answers
69 views

Factorial in power series; intuitive/combinatorial interpretation?

It is well known that the terms of the power series of exponential and trigonometric functions often involve the factorial function, essentially as a consequence of iterating the power rule. My ...
1
vote
2answers
33 views

Power series and $f(z) = z + f(z^2)$

Use power series $f(z) = \sum_{j=0}^\infty a_j z^j$ to solve the functional equation $f(z) = z + f(z^2)$. First, I'm not really sure what the question is asking for. Usually, when I see a ...
0
votes
0answers
8 views

sample size calculation for count data

I have a plan to see some treatment effect in several projects in my company where I will compare the average number of errors now and after the treatment. So what I know from current situation is ...
0
votes
1answer
22 views

Meromorphic Function on Extended Plane

How do I prove that every meromorphic function on the extended plane is a rational function?
0
votes
1answer
19 views

Coefficients of power series

After expansion, we have $$ (x_1+x_2+\dots+x_n)^m=a_1x_1^m+a_2x_1^{m-1}x_2+\dots $$ where $x_{()}$ is the variable and constant indices $n>m$. What is the expressions of all these possible ...
2
votes
1answer
27 views

Convergence of a power series for all $x \in \mathbb{C}$

I came across the following power series while looking at a problem. $$-1+a_1x+a_2x^2+a_3x^3+.......$$ where $a_0=-1$ and the choice of $a_1$ is arbitrary.The other coefficients are dependent on ...
1
vote
2answers
67 views

Possible counter example to theorem on Taylor series in complex domain

A well known theorem on Taylor series in complex domain is as follow: Suppose $f(z)$ has Taylor series at $a$ with convergence radius of $R$. Then $f(z)$ has at least one singular point on $|z-a|=R$. ...
2
votes
0answers
22 views

For $\sum_{n=1}^\infty (-1)^n\frac{(x-3)^n}n$ approximate when x = 4, with an error that does not exceed .01

For $\sum_{n=1}^\infty (-1)^n\frac{(x-3)^n}n$ approximate with specific details the series when x = 4, but with an error that does not exceed .01. That is, find a value of n so that the nth partial ...
1
vote
0answers
33 views

$k \pmod{n}$ functions

We are able to tell whether or not a function is even or odd based upon two tests; for example, an even function has the property that $$f(-z)=f(z)$$ and an odd function that $$f(-z)=-f(z)$$ It ...
0
votes
1answer
49 views

Integral of power series over a region

Let $$ F(x) = \sum_{n=0}^ \infty a_nx^n$$ where the power series converges in a neighborhood of the origin. Compute $$ \mu(F)= \sup \{ \delta > 0 : \text{there exist} \ \epsilon > 0 \ ...
1
vote
1answer
26 views

Manipulating a Power Series to change convergence at endpoints

The problem is to find a power series that satisfies convergence on the interval $[-1,3), (-1,3), (-1,3],$ and $[-1,3]$. I have worked out a general form equation that satisfies the first two, but I ...
4
votes
2answers
136 views

If $\sum_\limits{n\in\mathbb{N}}{b_n}$ is divergent, then so is $\sum_\limits{n\in\mathbb{N}}{\dfrac{b_n}{1+b_n}}$? [duplicate]

I want to prove the following: Suppose $\sum_\limits{n\in\mathbb{N}}{b_n}$ is divergent. Then $\sum_\limits{n\in\mathbb{N}}{\dfrac{b_n}{1+b_n}}$ is divergent. I think you have to prove by ...
0
votes
1answer
39 views

How to Evaluate this Series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}(1+2^n)}{n2^n}$?

I have to study complex series $$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}(1+2^n)}{n2^n}$$ and prove that series is equal to $\log(3)$. That series I have to stady in the circumference $C(0,R)$ and ...
0
votes
1answer
49 views

Taylor series and radius of convergence: $\sqrt{x}$ with centre $x = 16$?

I've been struggling with this question for a while now and getting nowhere with it. Could someone please help me out? Assuming that the function has a power series expansion about the given point, ...
0
votes
1answer
36 views

Standard technique for fiddling with power series

I will try a standard technique for fiddling with power series. If $g(t) =\sqrt{f(t)} $, then, differentiating, $g'(t) =\frac {f'(t)}{2\sqrt{f(t)}} =\frac {f'(t)}{2g(t)} $ so $2g'(t)g(t) = f'(t) $. ...
2
votes
1answer
40 views

Convergence of power series dependent on parameter

I want to prove that $$\sum_{n=1}^{\infty}{\frac{x}{n^{\alpha}(1+nx^2)}}$$ converges for each $x\in\mathbb{R}$ whenever $\alpha>\frac{1}{2}$. How can I prove this. Which test I have to use?
0
votes
1answer
57 views

How to solve $xy''+2y'+\lambda^2 \, xy=0$ with the power series method?

Find all functions $y(x)$ which can be expressed as a convergent power series $y(x)=\sum_{n=0}^{\infty} a_n x^n$ and which satisfy the following differential equation: $$xy''+2y'+\lambda^2 \, xy=0$$
0
votes
1answer
18 views

Discrete mathematics power series

I've hit a wall with this problem. So, i have been given: $A(x)=a0+a1*x+a2*x^2...ak*x^n$ and $B(x)=1/(1-x)*A(x)$ How do i show that $[x^n]*B(x)=sum(ak, k=0 .. n)$. I cant find a way through how ...
0
votes
1answer
31 views

radius of convergence of integral of power series

Suppose $\sum \limits_{n=0}^{\infty} a_n x^n$ has radius of convergence R. What is the radius of convergence of $\sum \limits_{n=0}^{\infty} \frac{a_n x^{n+1}}{n+1}$? How do I solve this without ...
1
vote
1answer
20 views

Find a power series representation for the function. (Assume $a > 0$.)

I'm down to my last attempt (my teacher allows $5$ tries per question)! Thank you!!
0
votes
1answer
22 views

The radius of convergence of a power series.

If I have a power series $$\sum_{j = 1}^{\infty}a_jx^{2j+1} = x\sum_{j = 1}^{\infty}a_jx^{2j} $$ Given that I have the radius of convergence $R$ of $$\sum_{j = 1}^{\infty}a_jz^{j}$$ where $z = ...
1
vote
0answers
47 views

Prove that the Puiseux series is algebraically closed

Denote by $K=\mathbb{C}((z))$ the fraction field of $\mathbb{C}[[z]]$. Define an embedding of $K$ onto itself taking $a(z)$ to $a(z^n)$ $\forall n$. The target is $\mathbb{C}((z^{1/n}))$. Define the ...
1
vote
1answer
32 views

Find a power series representation (centered at x = 0) and determine the radius and interval of convergence

For the following function: $f(x) = (x/(2-x))^3$ How do I find a power series representation (centered at x = 0) and determine the radius and interval of convergence? I managed to simplify the ...
1
vote
1answer
65 views

Calculating $\sum_{n=1}^\infty {\frac{nx^n}{4n^2-1}}$ [closed]

I would appreciate any help calculating the series. And determine where does the series converge uniformly. $$\sum_{n=1}^\infty {\frac{nx^n}{4n^2-1}} $$
0
votes
1answer
49 views

Inverse for $1-zb(z)$

I need to find the inverse of $1-zb(z)$ with $b(z)=\sum_{n=0}^{\infty}b_nz^n$. I have tried several approaches where I among other things have tried using the methods in my calculus book but nothing ...
0
votes
2answers
28 views

Power series expansion

I am trying to solve a equation which I have already solved using oDE but I want to solve it using a power series expansion but how do I express y as a power series? Equation is as below $$ (1+x) ...
0
votes
0answers
24 views

Reference request: attempt to give credit for power series result.

I've been informed that the result below is known by different names in different fields and I'm simply looking for the best person to credit. To my knowledge it originates with Kemp & Kemp ...
4
votes
5answers
275 views

Power series summation [closed]

Trying to find the sum of the following infinite series: $$ \displaystyle\sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{(2n-1)3^{n-1}}$$ Any ideas on how to find this sum?
1
vote
1answer
41 views

Different series representation for the same function

Ok, we know that $\frac{1}{1-x}=1+x+x^2+x^3+\cdots$. Now if we want to have a series representation of $y=\frac{1}{2-x}$ then there are two approaches: First ...