0
votes
2answers
24 views

Intervel of Convergence of a Power Series

Can anyone explain how to do this problem? I think you might be able to approach it with the ration test but I'm unsure. Any help is greatly appreciated! $$\sum_{n=0}^{\infty} \frac{(2x-3)^n}{n \ ...
1
vote
1answer
61 views

Does the series $\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ converge?

$\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ Please let me know how you did it. Thank you.
0
votes
2answers
41 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 ...
0
votes
0answers
17 views

conflictions of analytic functions to the boundary and Schwarz reflection principle

Let $\Omega$ be an open subset of $\mathbb{C}$ and $f:\Omega\longrightarrow \mathbb{C}$ be a holomorphic function. Then for any $z\in \Omega$ and any $r>0$ such that $D(z,r)\subseteq \Omega$, $f$ ...
1
vote
1answer
28 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 ...
1
vote
2answers
34 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: $$ \lim_{n\to\infty}\big|\frac{64^{n+1}(x-10)^{3n+4}}{64^n(x-10)^{3n+1}}\big| $$ ...
1
vote
1answer
23 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 ...
0
votes
3answers
38 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!
1
vote
2answers
20 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 ...
2
votes
1answer
46 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 ...
3
votes
3answers
118 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 ...
3
votes
2answers
50 views

Finding terms of a Taylor series where $f(x)$ is a function with a power

I've been stuck with this Taylor series problem for a while now. We have that $$ f(x) = (1 + x^2)^{-2/3} $$ and it's centered at $0$. So what I thought of doing was the $$ \frac{f^{n}(a)(x - ...
0
votes
1answer
28 views

What is the sum of this series: $\sum_{k=n}^{\infty}(1-p)^{k-2}\cdot p$?

I have: $$\sum_{k=n}^{\infty}(1-p)^{k-2}\cdot p$$ $|p|<1$. The sum is: $$=\frac{(1-p)^{n-2}\cdot p}{1-(1-p)}=(1-p)^{n-2}$$ Or I wrong? I use the fact that the sum of a series is: $\frac{a}{1-p}$ ...
1
vote
3answers
191 views

Find complicated Taylor Series

According to some software, the power series of the expression, $$\frac{1}{2} \sqrt{-1+\sqrt{1+8 x}}$$ around $x=0$ is $$\sqrt{x}-x^{3/2}+\mathcal{O}(x^{5/2}).$$ When I try to do it I find that I ...
0
votes
0answers
30 views

Reference for power series

I would need some references for power series, Taylors series of elementary functions, derivation and integration of power series, convergence of sequences of functions and series of functions. The ...
1
vote
2answers
38 views

Simplifying ratio test with exponents $k+1$

Question: Find the interval and radius of convergence. $$\sum_{k=1}^\infty\frac{(x-1)^k(k^k)}{(k+1)^k} .$$ I applied the ratio test. ...
0
votes
1answer
106 views

If $f(2x)=2xf'(x)$, then find $f(x)$

If $f(x)$ is Analytic functions on $R$,and such $$2xf'(x)=f(2x)$$ Find all $f(x)$ My idea: let $$f(x)=\sum_{n=0}^{\infty}a_{n}x^n$$ so I can't Thank you
2
votes
1answer
33 views

Power series centered at $x =0$

I have this question in my advanced calculus textbook. Give an example of a power series centered at $x = 0$ that converges on $[-2,2]$ but not absolutely on the entire interval $[-2,2]$, and ...
8
votes
1answer
130 views

Taylor Series of $\frac{1}{1-\cos x}$

The problem is, as the title suggests, to find the Power Series Expansion of $\frac{1}{1- \cos x}$ around $x=c$. What I've tried: Direct Computation: Derivatives get very ugly quickly, and don't ...
1
vote
0answers
38 views

Convergence set of power series

I am trying to find the convergence set of the power series: $\sum_{n=1}^\infty ln\big[1+\big(\dfrac{1}{n}\big)\big](x+2)^n$. So using the ratio test: $\lim_{n\to\infty} \dfrac{|a_{n+1}|}{|a_n|} = ...
0
votes
3answers
119 views

How to find the sum of this power series $\sum\limits_{n=0}^\infty \frac {x^{5n}} {(5n)!}$

How to prove that $$ \sum\limits_{n=0}^\infty \frac {x^{5n}} {(5n)!}= \frac{2}{5} e^{-\cos \left( 1/5\,\pi \right) x}\cos \left( \sin \left( 1/5\,\pi \right) x \right) +\frac{2}{5}\, e^{\cos ...
3
votes
2answers
74 views

If it converges, how to show that power series converges to $f(x)$?

I had a very basic question. Suppose $f(x)$ is a function. And let us say it has a power series :- $$f(x) = \sum_{n=0}^\infty a_nx^n.$$ Suppose we are operating inside the region of convergence. ...
0
votes
2answers
89 views

Prove that $f(z) = \sum\limits_{k = 1}^\infty \frac{z^{2^k}}{2^k}$ is continuous in the closed unit disc and holomorphic inside it.

I have started off by assuming that there is a disc of radius $r$ for which $|z|<r$ for $r \in (0,1)$ and $z \in D_r$. This implies that $|z|^{2^k} < r^{2^k}$ And after that, I don't know ...
5
votes
4answers
109 views

Maclaurin Series of $\frac{1}{e^x -1}$

I want to find the Maclaurin Series for the function $f(x) = \frac{1}{e^x -1}$. But when I compute the first derivative of f(x): $$ \frac{d}{dx}\frac{1}{e^x -1} = -\frac{e^x}{(e^x-1)^2} $$ A the ...
1
vote
1answer
34 views

Logarithm expansion

I have a problem showing that the following identity for power series holds true: $$ \ln(1+x+x^2+x^3+...)=\sum_{n\geq1}\frac{x^n}{n} $$ when $\left|x\right|<1$. Can anyone help me, please? ...
0
votes
1answer
27 views

Closed form of a Mac Laurin Series expansion [closed]

I am looking for a closed form for the MacLaurin Polynomial for $$ \exp\left(\frac{x^n}{n}\right) $$ where $n\in\mathbb{N}$. Thanks in advance for your help Kijn.
1
vote
1answer
93 views

Taylor series of $\frac 1 {1+x^2}$

I have to construct the Taylor series of $$\frac 1 {1+x^2}$$ around $0$ and $1$ and analyze the convergence in both cases. Also (but this is a consequence of the previous series) I have to construct ...
0
votes
2answers
111 views

How to turn $-\ln(1-x^2)$ into a power series representation?

I need to turn $f(x)=-\ln(1-x^2)$ into a power series, there are 2 things I can't understand: 1) I don't how to check and even if to check whether this function can turn into a power series or not, ...
1
vote
2answers
174 views

For which $x$ does the series $∑_{n=1}^∞(1+\frac{1}{2}+ \frac{1}{3}+⋯+\frac{1}{n})\, x^n$ converge?

Determine for what value of $x$ the series converges $$\sum_{n=1}^\infty \left(1+\frac{1}{2}+ \frac{1}{3}+⋯+\frac{1}{n}\right) x^n $$ Observe that $∑_{n=1}^∞(1+\frac{1}{2}+ ...
0
votes
1answer
53 views

convergence of series, double factorial against power function?

It is known that the series $$\sum_{n=1}^{\infty} \frac{C^n}{n!}<\infty$$ for any $C>0$, that is, the factorial kills the Power function. I wonder now if $$\sum_{n=1}^{\infty} ...
0
votes
1answer
41 views

determine the convergence region of a complex series

Determine the region $\Omega$ of the complex plane such that for any $z\in\Omega$ the following series converges: $\sum_{n=1}^\infty\frac{1}{n^2}\exp(\frac{nz}{z-2})$. I do not know how to treat ...
0
votes
2answers
30 views

Find the Maclaurin series of $f(x)=\frac{x}{x^4+x^2+1}$ [closed]

Maclaurin series for $f(x)$. Thanks.
0
votes
1answer
54 views

Integrating a Taylor series term-by-term

Why is $$\int_{0}^{z} \frac{\sin x}{x} \ dx =\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)!} \int_{0}^{z} x^{2n} \ dx$$ not valid for $z= \infty$? Well, at least I'm assuming it's not valid since ...
0
votes
1answer
25 views

Series expansion of quotients

I'd like to start of with a simple formula from a textbook $T(W) = \frac{W^5}{1-2W} = W^5 + 2W^6 + 4 W^7 + \dots + 2^j W^{j+5} + \dots$ Obviously, this is an expansion of the quotient into a power ...
2
votes
1answer
49 views

Clever way to expand 1/(z^2-n^2) in power series?

Is there a good trick to prove the following identity? $$\frac 1 {z^2-n^2} = -\sum_{i=0}^\infty \frac {z^{2i}} {n^{2(i+1)}}$$ I tried writing out the coefficients as a Taylor series, but this was ...
2
votes
2answers
51 views

How can i evaluate this power series?

$\sum_{n=0}^{\infty }\frac{1}{2n+1} \left (\frac{1}{3} \right )^{n}\left ( -1 \right )^{n} $ it's solved by power series of arctan. is it possible the answer written by real number?
0
votes
2answers
43 views

how can I sove approximation evaluation of this integral?

$$\int_{-1}^{0}\sin(e^{x})\,dx $$ approximation of this formula up to difference(error) $1/5000$ Because of the error size $1/5000$ , I think it's solved by taylor expansion.
0
votes
2answers
45 views

$ \sum_{n\geq1}(\frac{1}{(n+1)!} ∏_{k=1}^{n} f(k))$ converge or diverge?

Let $f: \mathbb N - {0} \to \mathbb N -{0}$ injective function, verify is the serie $ \sum_{n\geq1}(\frac{1}{(n+1)!} ∏_{k=1}^{n} f(k))$ converge or diverge . I prove for $n=5$ and $n=9$ and found ...
2
votes
1answer
68 views

Find the sum of $\sum_{n=1}^\infty \frac{x^{n-1}}{3^nn}$ - What is wrong with my solution?

I have to find the sum of the following power series: NOTE: please assume that x is in the convergence domain. $$\sum_{n=1}^\infty \frac{x^{n-1}}{3^nn}$$ My ...
4
votes
2answers
76 views

Series of inverses of binomial coefficients

Can you think of a simple way of proving that $$ \sum_{n=k+1}^\infty \frac{1}{n \choose k} $$ is rational for any $k \geq 2$? Here's the background. Consider a series: $$ \sum_{n=1}^\infty ...
1
vote
2answers
43 views

convergence ratio of the serie $e^{xn}$

How can I determine the values of $x$ such that the series converge: $$\sum_{n=0}^\infty e^{xn}$$ I'm really lost in this problem, please help.
1
vote
2answers
130 views

Converge of the sum $\sum_{k=1}^{n} k x^k $

For what values ​​of x the sum converges and what is the limit when $n \rightarrow \infty$ $\sum_{k=1}^{n} k x^k $ My work: First i try to calculate the interval and radius of convergence of ...
2
votes
2answers
79 views

Expansion and convergence of $\sum_{m=1}^{\infty}\frac{\sin(2\pi n x^{1/m})}{ m}$

Consider the series: $$\sum_{m=1}^{\infty}\frac{\sin(2\pi n x^{1/m})}{ m}\;\;\;\;n\in\mathbb{N}$$ Other than formal manipulation of the Taylor series of the $\sin$ function, is there a way to expand ...
3
votes
1answer
52 views

Analytic function satisfying $x^2f''(x) +xf'(x)+x^2f(x)=0$ and $f(0)=1$

Suppose that the power series $\sum_{n=0}^\infty a_nx^n$ converges for all real $x$ to a function $f(x)$ that satisfies $$x^2f''(x) +xf'(x)+x^2f(x)=0 \quad \text{and} \quad f(0)=1.$$ ...
2
votes
2answers
63 views

Formal power series with all derivatives zero

I have the following question. Suppose I have a formal power series $f(x)=\sum\limits_{i=0}^\infty c_ix^i$ with real coefficients. Suppose that all the derivatives ...
20
votes
1answer
243 views

Is it possible for a function to be smooth everywhere, analytic nowhere, yet Taylor series at any point converges in a nonzero radius?

It is well-known that the function $$f(x) = \begin{cases} e^{-1/x^2}, \mbox{if } x \ne 0 \\ 0, \mbox{if } x = 0\end{cases}$$ is smooth everywhere, yet not analytic at $x = 0$. In particular, its ...
5
votes
1answer
101 views

Simplify $\sum_{n=0}^{N}\binom{N}{n} \frac{a^{N-n}}{n!} \frac{d^n}{dx^n} f(x)$

Simplify the following expression $$S_N = \sum_{n=0}^{N}\binom{N}{n} \frac{a^{N-n}}{n!} \frac{d^n}{dx^n} f(x), $$ where $a$ is a real number and $f(x)$ is an analytic real function. What is $\lim_n ...
12
votes
2answers
291 views

Finding the derivative of $x\uparrow\uparrow n$

I am trying to find a general derivative for the function: $f(x)=x^{x^{x^{...^{x}}}}$however to do that I must find $f^{\prime }$ and $f^{\prime \prime}$...etc. I am now trying to write down a general ...
4
votes
2answers
101 views

Maclaurin expansion of arctan: convergence?

In my textbook, the Maclaurin series expansion of $\arctan{x}$ is found by integrating a geometric series, that is, by noting that $\frac{d}{dx}(\arctan(x)) = \frac{1}{x^2+1}$ then rewriting the ...
1
vote
1answer
46 views

Exact expression for series

Can the exact expression for the following series be found, given $|x|<1$? Just curious. $f(x) = \frac{x^2}{17}+\frac{x^3}{3}+\frac{x^4}{3}+\frac{x^5}{3}+ \ldots$