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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1answer
23 views

Explanation on how to turn a numerical sequence into a power serie

I'm taking a calculus class, but I skipped school the past week due to health problems. I spoke to my teacher and classmates and they told me that they had seen power series topic. So I got a copy of ...
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0answers
57 views

Test the uniform convergence of the series in indicated region

Test the uniform convergence of the series I tried to find $M_n$ such that $|\sum_{n=1}^ \infty(-1)^n\frac{z^{2n-1}}{1-z^{2n-1}}|\le M_n $ by using Abel's Theorem This is the question : Test the ...
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1answer
151 views

Taylor series convergence with natural logs

I am working on this problem. Find Taylor series of function $f(x)=\ln(x)$ at $a = 6$. $$f(x) =\sum_{n=0}^\infty c_n (x- 6)^n$$ I seem to be having trouble with the interval of convergence can ...
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2answers
124 views

how to multiply infinite power series

I am doing an assignment for my precalculus 2 class. I am expanding two infinite power series and multiplying them together to prove that $\exp(ax)\exp(by) = \exp(ax+by)$ I'm not sure what I am ...
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1answer
199 views

Differential equation by series solution method: equating coefficients to zero

I am following the solution for a problem, and I am stuck at the following equation: $$2a_2+\sum_{n=1}^\infty \left[(n+2)(n+1)a_{n+2}-a_{n-1}\right]x^n=0\tag1$$ Now, the professor equates the ...
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1answer
49 views

How to justify, $\sum_{n=1}^{\infty} a_{n} x^{n} - \sum_{n=1}^{\infty}a_{n}y^{n}=\sum_{n=1}^{\infty} a_{n} (x^{n}-y^{n})$?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$ and we let $K_{1}$ be a compact subset of ...
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1answer
36 views

What is the significance of finding the series solution of a differential equation “about a point”?

I am learning the series solution method of solving differential equations, and I am curious as to what the rationale is for finding out the solution of the equation about a particular point. It seems ...
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2answers
167 views

Power series with $f(x)=\frac {1}{1+100x^2}$

I am working on the power series. Here is the question $$f(x)=\frac {9}{1+100x^2}$$ represented as a power series $$f(x) = \sum^{\infty}_{n=0}c_nx^n$$ I need to find $c_0,c_1,c_2,c_3,c_4,R$ I got ...
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2answers
55 views

Is it true that, $|e^{x}-e^{y}|\leq C \cdot |x-y|$?

Define $f:\mathbb R \to \mathbb R$ such that $f(x)= e^{x}-1:= \sum_{n=1}^{\infty} \frac{x^{n}}{n!};$ for $x\in \mathbb R.$ My Question: Can we expect $|f(x)-f(y)|\leq |x-y| \cdot C;$ where $C$ is ...
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2answers
112 views

Finding Function of Series: $e^{-kx}$

If the series representation of $e^{-x}$ is: $$\sum_{k=0}^{\infty} \frac{(-x)^k}{k!} $$ Then what is for $e^{-kx}$?
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0answers
235 views

Question about Big O notation for asymptotic behavior in convergent power series

Examples of such use of Big O notation can be found for instance on Wolfram Alpha here. More details on the Wikipedia page. The idea, as I understand it, is that the term between parenthesis in Big O ...
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0answers
39 views

Question about Big O notation for asymptotic behavior in convergent power series [duplicate]

Examples of such use of Big O notation can be found for instance on Wolfram Alpha here. More details on the Wikipedia page. The idea, as I understand it, is that the term between parenthesis in Big O ...
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1answer
41 views

Coefficient of power series when $p(x) = \sum b_nx^n$ converges for $|x| \le 1$ and $p(x) = 0$ for $|x| \lt \delta$.

Suppose that the power series $p(x) = \sum b_nx^n$ converges for $|x| \le 1$. Suppose that for some $\delta \gt 0 , p(x) = 0$ for $|x| \lt \delta$. Show that $b_n = 0$ for all $n \ge 1$.
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3answers
34 views

Don't know why this power series representation is wrong…

I've run into something confusing. The problem is that I have to find the power series representation of $g(x)$ using the given $f(x)$, specifically $g(x) = \ln(1 - 3x)$ using $f(x) = \frac{1}{1-x}$. ...
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3answers
283 views

if $ S(x)=\sum_{n=0}^{\infty}a_{n}x^n,|x|<R$, $S_k(R)$ bounded,prove or disprove $\lim_{x\to R}S(x)$ exist?

let $$S(x)=\sum_{n=0}^{\infty}a_{n}x^n,a_{n}>0,|x|<R$$ (or mean that powr series have radius of convergence R.) and let ...
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1answer
19 views

A integral about Powers of x and binomials.

Here is the integral $$\int_0^\infty {\frac{{{x^{p - 1}}}}{{x + a}}{{\left( {bx + c} \right)}^q}} dx,where{\text{ }}a,b,c > 0,p,q \geqslant \frac{1}{2}$$
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1answer
26 views

Radius of convergence of $\sum a_nx^n$ where $a_n = {k \choose n}$

Consider the power series $\sum a_n x^n$ where $$ a_n = {k \choose n} $$ for some $k$. What is the radius of convergence of this power series? I got one. Does that seem correct? I got that the ...
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2answers
279 views

What is the Taylor series of $\frac{1}{\sin(z)}$ about $z_0 = 1$?

This was a exam question so I know it cannot take too long to write out the proof. Only I cannot see an answer. I would imagine you write $\sin(z) = \sin(1+(z-1)) = \sin(1)\cos(z-1) + ...
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5answers
79 views

Finding sum of Power series

Hi could anyone help me with this question Determine the sum of the power series: $$S=-\sum_{n=1}^{\infty}\frac{(1-x)^n}{n}$$ Where x=1.74 I tried to differentiate this expression, but I do not ...
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1answer
158 views

Determine the value of r where the series converges

show that $$ \big(r\big)^{ln(n)} = \big(n\big)^{ln(r)} $$ Then determine the values of r (with r>0) for which the series $$ \sum_1^\infty (\big(r\big)^{ln(n)})$$ converges. r must be in what ...
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1answer
57 views

Absolute and conditional convergence of a series with $\sin(x)$

I have to explore absolute and conditional convergence of this function series I tried to find $a(n)$ and $a(n+1)$ terms of the series and then divide it and take a limit. But I've got nothing. ...
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3answers
63 views

What is the sum notation equivalent to this series?

I tried to solve this question Find radius of convergence of the series $$1+ \frac{a.b}{1.c}z + \frac{a(a+1)b(b+1)}{1.2...c(c+1)}z^2+.... $$ The book answer is 1 I cant obtain the nth term in the ...
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2answers
288 views

find region of convergence of the series

I tried to solve this question Find domains of convergence of the series $$\sum_{n=1}^ \infty \frac{z^n}{n (\log n)^2 }$$ How can I do this .
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2answers
31 views

Find domains of convergence of the series

I tried to solve this question but the final solution which I obtain is not the as same as in the text book Find domains of convergence of the series $$\sum_{n=1}^ \infty ...
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2answers
31 views

Calculate ch(0.2) to the nearest 0.01

Help me calculate ch(0.2) to the nearest 0.01. I tried to rewrite ch as a series but I still don't know how to evaluate it and what to do with factorial Help me please. it's very important
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3answers
104 views

Find a radius of convergence of power series

I have to Find a radius of convergence of this power series I' ve decided to use D'alambert indication: Looking for a limit i meet a problem with a factorial Please. help me finish this ...
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1answer
45 views

Power Series to solve non linear differential equations.

I've been revising Power series recently and their application when it comes to solving linear differential equations, but in this question I'm not sure what to do when it's a non linear function. I ...
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1answer
42 views

Express the coefficients of $\exp(f(x))$ using the coefficients of $f(x)$

Given a formal series $$f(x)=\sum_{k=1}^\infty f_k x^k$$ what is $$K_n:=\left[\left(\frac{d}{dx}\right)^n e^{f(x)}\right]_{x=0}$$ in terms of the coefficients $\{f_k\}$? I stumbled upon this ...
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0answers
44 views

Replacing $q^2$ by $q$

I have a rather strange question. Suppose we are given a formal power series $$S(q^2) = \sum_{n = 0}^\infty a_n q^{2n}.$$ I wish to replace $q^2$ by $q$. This implies that $S(q) = \sum_{n = 0}^\infty ...
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2answers
240 views

What is $\sum_{n=0}^{\infty}|a_nz^n|^2=\frac{1}{2 \pi}\int_{-\pi}^{\pi}|f(ze^{it})|^2dt$ for?

Let $f : \mathbb C\rightarrow \mathbb C$ be an analytic function : $f(z)= \sum a_n z^n$ It holds that $$a_n z^n= \frac{1}{2 \pi}\int_{-\pi}^{\pi}f(ze^{it})e^{-int}dt$$ and ...
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2answers
29 views

Find a power series by comparing it to a geometric series?

Find the power series for the following function: $f(x)=\frac{1-x}{x-3}$ centered at x=1. This is what I've done: ...
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2answers
39 views

Question about Power Series

I have this problem: $$\sum_{n=0}^{\infty}\ 64^n\ (x-10)^{3n+1}$$ After using the Ratio Test I am left with this: $$ \lim_{n\to\infty}\big|\frac{64^{n+1}(x-10)^{3n+4}}{64^n(x-10)^{3n+1}}\big| $$ ...
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2answers
43 views

Problem understanding notation

I'm learning about generating functions and in the opening explanations my book (and various sources) claim: $$a_n = 1 \forall n \in \mathbb{N}_0, \ \ \ f(x) = \frac{1}{1-x}$$. I read this as: ...
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2answers
72 views

Interval of Converge for a Power Series

I would really appreciate some help on this problem that I have been working on. It's a power series question. I have to find the interval of convergence for this power series. The power series is: ...
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2answers
53 views

proving series convergence by definition

I have a question in the homework, and I tried to prove it but I don't know if it is correct? I want to prove that the series $$ \sum_{n=1}^{\infty} \frac{z^n}{n(n+1)} $$ is absolutely convergent for ...
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2answers
180 views

Find the power series for $f(x) = \frac{\cos(x^3)}{2x^2}$

I'm pretty sure if it were just $\cos(x^3)$ i could subsititue $x^3$ for $x$, everywhere in the known series, but what do I do because it's divided by $2x^2$?
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2answers
83 views

Is there any closed form for this series?

It's a power series that I found during the computation for my research. \begin{equation*} \sum_{k=0}^n \binom{n}{k}\frac{n!}{(n-k)!}x^{n-k}(-1)^k. \end{equation*} Without the annoying term of ...
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1answer
109 views

Nth Term Proof Taylor Polynomial

suppose that Msub_n(x) is the nth order maclaurin polynomial for f(x). Show that if k is a constant then Msub_n(kx) is the nth-order maclaurin polynomial for f(kx)
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2answers
50 views

Let $f(z) = \sum_{j=0}^{\infty}a_jz^j$ be the Maclaurin expansion of a fnction $f(z)$ analytic at the origin. Prove each of the following statements.

Let $f(z) = \sum_{j=0}^{\infty}a_jz^j$ be the Maclaurin expansion of a function $f(z)$ analytic at the origin. Prove each of the following statements. $(a)$ $\sum_{j=0}^{\infty}a_jz^{2j}$ is the ...
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1answer
80 views

Find the power series for $d/dx(\arcsin x)$

How would you find the general power series for $\frac{1}{\sqrt{1-x^2}}$ , without using the general rule for arcsinx? I understand it is necessary to use binomial series, but I am having trouble ...
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0answers
53 views

Solution regarding Power Series and ODE's

About 4 months ago I posted Series solution to $y''-xy'-y=0$. I ran through the analysis and it appeared that I solved the ODE . The solution seemed to be ...
2
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2answers
64 views

Radius of convergence two power series (by using Cauchy test).

Let power series $\sum_{n=0}^{\infty} a_nz^n$ have radius of convergence $R$. I would ask you, is it true that $\sqrt[n]{a_n} \rightarrow \frac{1}{R}$? If it is true, then power series ...
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1answer
50 views

Power series of $f(z) = \frac{z}{1-z}$

Find power series of $f(z) = \frac{z}{1-z}$ in point $z_0 = i$ and find radius of convergence this power series. Of course, I can find $f^{(n)}(z_0)$ and then I will have $$f(z) = \sum_{n=0}^{\infty} ...
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3answers
47 views

Why $\sum_{k=1}^n (\frac56)^{k-1}\cdot (\frac16)=1-(\frac56)^n$

Why $$\sum_{k=1}^n \left(\frac56\right)^{k-1}\cdot \left(\frac16\right)=1-\left(\frac56\right)^n$$?? Which formula I need to use to calculate it? Thank you!
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1answer
29 views

Maclaurin series and general terms

For a general term in a maclaurin (or any other) series e.g. x^r/r! is it always that r=0, or r=1 at the first term or does it have nothing to do with the term it appears in e.g. r=10 could come ...
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2answers
97 views

Represent Power Series of a function

Hi could anyone help me answer this question Find the power series representation for the function and determine the radius of convergence $f(x)=\frac{x^2}{\left(1-2x\right)^2}$ After getting ...
2
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2answers
102 views

Power series of a function about a non zero point

No clue how to ask questions here so here goes nothing! How do I work towards finding the power series of a function centered about a point a not equal to $0$? The specific question I was asked is to ...
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2answers
232 views

Finding the Function of a Power Series: $\sum kx^{k+1}/3^k$

Given: $$\sum_{k=1}^{\infty} \frac{kx^{k+1}}{3^k}$$ Im guessing its equivalent to: $$\sum_{k=1}^{\infty} k\left(\frac{-1}{3}\right)^k x^{k+1}$$ But I am not sure on how to advance past this step. ...
2
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1answer
51 views

Convergence of a series that looks similar to $e^x$

Suppose I have some $\epsilon > 0$ and some constant $c > 0$. Does the series $$ \sum_{n=1}^{\infty} \frac{c^{n^{\epsilon}} }{[n^{\epsilon}]!}, $$ where $[r]$ is the integral part of a real ...
5
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3answers
298 views

$\sin^2(x)+\cos^2(x) = 1$ using power series

In an example I had to prove that $\sin^2(x)+\cos^2(x)=1$ which is fairly easy using the unit circle. My teacher then asked me to show the same thing using the following power ...