Questions tagged [power-series]
Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n (x-c)^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex.
1
vote
3answers
36 views
Power series representation for $x\arctan(x)$. How?
I am supposed to start with the P.S representation for $\dfrac{1}{1-x}$ and then figure out the P.S representation for $x\tan^{-1}(x)$. I know the process of making it for just the $\tan^{-1}x$. But ...
0
votes
1answer
29 views
Combining geometric series
I’m wondering how one would go about combining
$$\sum_{n = 0}^{\infty}z^{2n+1} - 4 \sum_{n = 0}^{\infty}z^{2n-1}$$
into one series?
0
votes
2answers
16 views
Residue of a product of series
I need to find the residue of $f(z)=\frac{e^{\frac{1}{z}}}{z-1}$ in $z=0$.
To do this, I proceeded to find the Laurent series of $f$ which is:
$\sum_{n=0}^{\infty} \frac{z^{-n}}{n!}\sum_{k=0}^{\...
0
votes
0answers
25 views
Prove that the function $f(x) = \sum_{k =1}^{\infty} \frac{\sin(kx)}{2^{k}}$ is infinitely differentiable
Prove that the function $f(x) = \sum_{k =1}^{\infty}
\frac{\sin(kx)}{2^{k}}$ is infinitely differentiable
This is a practice problem for an exam I have coming soon. I am trying to study, but I ...
2
votes
0answers
41 views
Preimage of a prime ideal in a ring of formal power series.
Let $k = \overline{k}$ be a field. We have the inclusion $\iota:k[x,y] \to k[\![x,y]\!]$ and the prime ideal $\mathfrak{p} = (y - \sum_{n\geq 1} x^n/n!)$. This ideal is prime, because it is the kernel ...
0
votes
0answers
37 views
Radius of convergence of $\sum \frac{n!}{n^n} a_n z^n$
I'm trying to prove that the radius of convergence of $\sum \frac{n!}{n^n} a_n z^n$ is $eR$ when $R>0$ is the radius of convergence of $\sum a_n z^n$.
I can easily obtain that the radius of ...
1
vote
1answer
21 views
An even function, $f(z)$, analytic near 0 can be written as another analytic function, $h(z^2)$
If f is an even function, $f(z) = f(-z)$, and is analytic near 0, then there exists a function h, also analytic near 0, such that $f(z) = h(z^2)$
I suspect this statement is true because the ...
1
vote
2answers
35 views
How can I show $\sum_{k=0}^{\infty} \frac{1}{1 + |2|^{k}}$ converges? [on hold]
How can I show $\sum_{k=0}^{\infty} \frac{1}{1 + |2|^{k}}$ converges? One way would be to just compute it, but I don't know how to deal with the absolute value. Additionally, I want to show that $\...
0
votes
1answer
20 views
Radius of convergence of two power series and the sum of the power serieses
$R_1$ Is the Radius of convergence of $\sum_{k=1}^{\infty}a_nx^n$.
$R_2$ Is the Radius of convergence of $\sum_{k=1}^{\infty}b_nx^n$.
What can we say of the Radius of convergence of $\sum_{k=1}^{\...
0
votes
1answer
48 views
Calculate Sum function $na_n=\frac{2}{3}a_{n-1}-(n-1)a_{n-1}$
$a_0=3$, $a_1=5$, for arbitrary $n>1$ , $\displaystyle
na_n=\frac{2}{3}a_{n-1}-(n-1)a_{n-1}$
Proof: when $|x|<1$, $\displaystyle\sum_{n=0}^\infty{a_n}{x^n}$
convergence, and calculate ...
1
vote
2answers
67 views
Solution of $y''+e^xy=0$ is unbounded as $x\to\infty$
Consider the differential equation $y''+e^xy=0$. Can we say something about the behaviour of $y$ as $x\to\infty$? In particular, is it unbounded?
I think, to solve the equation, we need to use the ...
1
vote
1answer
29 views
Given $\lim_{n\to\infty} |a_{n}|^{1/n} = \alpha$, prove the following properties.
Given $\lim_{n\to\infty} |a_{n}|^{1/n} = \alpha$, prove:
1) If $\alpha > 0$, show $\sum_{n = 0}^{\infty} a_{n}x^{n}$ converges
if $|x| < 1/\alpha$ and diverges if $|x| > 1/\alpha$
...
2
votes
1answer
27 views
Existence of polynomial p such that $|f(x) − p(x^2)| < \epsilon$
Let $f$ be a real valued continuous function on $\left[−1, 1\right]$ such that $f(x) = f(−x)$ for all $x \in \left[−1, 1\right]$.
Show that for every $\epsilon > 0$ there is a polynomial $p\...
-1
votes
1answer
31 views
Using the power series of $\sin x^3$, the value of $f^{(15)}(0)$ is equal to $k\cdot11!$. Find the value of $k$.
I have the following question:
Using the power series of $\sin x^3$, the value of $f^{(15)}(0)$ is equal to $k\cdot11!$. Find the value of $k$.
I tried to write the power series using the one ...
0
votes
1answer
27 views
Find first five non-zero terms of power series
I have the function f(x) = $\dfrac{2x}{\left(x-5\right)^2}$, and I'm supposed to "find the first five non-zero terms of power series representation centered at x = 0."
Using $\frac{f^{(n)}(0)}{n!}$, ...
1
vote
1answer
57 views
Solve $2y''+y'=0$ using power series
I got the answer $C_0e^{-x/2}$ by doing
$$\sum_{n=1}^{\infty }\:\left(x^{n-1}\right) \left(2n\left(n+1\right)C_{n+1}\:\right)+\left(nC_n\right) = 0$$
thus giving
$$C_{n+1}= -\dfrac{C_n}{2(n+1)}.$$
...
0
votes
3answers
55 views
Explain $y''-y=0$
I've solved this equation and got $c_0\cosh x + c_1\sinh x$. However, I've noticed that if the arbitrary constant $c_0$ doesn't equal $c_1$, this wouldn't work; you would have to convert the $\cosh x $...
0
votes
0answers
28 views
Taylor series of a function
I am puzzled with the following problem and I am not able to figure out the answer no matter how hard I try. Let's say we have a function $f(x)$ and we know its Taylor series is of the form $\sum_{n=0}...
0
votes
1answer
20 views
Writing power series as function
I need to write the following power series as a function:
$2\sum_{n=0}^\infty (-1)^n x^{2n+1}$
I am not sure how to go about doing this.
1
vote
2answers
18 views
Write power series as rational function
I need to write the power series:
$\sum_{n=1}^\infty \frac{1}{(x-3)^{2n-1}} - \frac{1}{(x-2)^{2n-1}}$
I need to write it as a rational function. I am not sure how to go about doing this.
1
vote
2answers
28 views
Finding the first non-zero terms of a power series
I have the function:
$f(x) = \frac{30}{(x^2 + 1)(x^2-9)}$
I need to find the first four non-zero terms of the power series centered at zero. I have not had much experience with power series so I am ...
-3
votes
0answers
46 views
Why isn't the first term of the McLaurin series for $\cos(x)$ a pole? [closed]
As I understand it, the McLaurin series for $\cos(x)$ is
$$\sum_{k=0}^\infty \frac{(-1)^k x^{2k}}{(2k)!}$$
This leaves me puzzled. Websites I've seen give the expansion of the sum as
$$1-\frac{x^2}{...
-1
votes
2answers
58 views
Find the sum $\sum_{n=1}^{\infty} \frac{x^{n+1}}{(n+1)*(n+2)}$
Need to find the sum $\sum_{n=1}^{\infty} \frac{x^{n+1}}{(n+1)(n+2)}$
What I did based on the suggestions:
Multiplied and divided by $x$
"forgot" the $x$ in the denominator and then took the ...
2
votes
2answers
47 views
Confusion about radius of convergence of a power series
I'm a bit confused about the following statement from a script. We look at the power series with coefficient $ a_k = (1+1/k)^{k^2}$ and want to compute the radius of convergence $R$ of the series. The ...
-3
votes
0answers
19 views
0
votes
1answer
28 views
Proof area of {$(x,y) R^2 | 0 \le y < e^x$ and $0\le x< h$} is $e^h-1$
I'm supposed to prove that the area of {$(x,y) R^2 | 0 \le y < e^x$ and $0\le x< h$} is $e^h-1$
I was going to try to make it a function and calculate it using a Riemanns sum.
That led me to
...
9
votes
1answer
87 views
Seeing that $\lim_{x \to \infty} \sum (-x)^n/n! = 0$
Is there any way to see directly from the power series that $$\lim_{x \to \infty} \sum_{n=0}^\infty \frac{(-x)^n}{n!} = 0$$? I realize that $\displaystyle{\lim_{x \to \infty} e^{-x} = 0}$. That's ...
3
votes
2answers
46 views
Textbook Proposition on Product of Real Analytic Functions
Let
\begin{align*}
\sum\limits_{j=0}^\infty a_j (x-c)^j && \sum\limits_{j=0}^\infty b_j (x-c)^j \\
\end{align*}
be two power series with intervals of convergence $\mathcal{C}_1$ and $\...
1
vote
3answers
48 views
Radius of convergence of $\sum\limits_{n=1}^\infty((\frac{1}{4})^n+(\frac{1}{3})^n)x^n$
according to wolfram alpha the radius is 3. I'm struggeling with the proof and would be very glad if someone could take a short look. Here my approach using the root test:
$a_n:=\sqrt[n]{(\frac{1}{4})...
-3
votes
1answer
39 views
Sum of series $\sum\limits_{k=0}^\infty \frac{a^{k^2}}{k!}$
I am looking at this sum:
$\displaystyle\sum_{k=0}^\infty \frac{a^{(k^2)}}{k!}$ for some $a>0$.
3
votes
0answers
45 views
an inverse of the Artin-Hasse exponential?
In the p-adic world the Artin-Hasse exponential is the sollowing power series:
$$
E_p(x)= exp \left( \sum_{n=0}^{\infty}\frac{x^{p^n}}{p^n} \right)
$$
where $E_p(x)\in 1+x\mathbb{Z}_{(p)}[[x]]$ with ...
0
votes
1answer
24 views
Find the expectation of this mass function
$p_x$ is defined as follows:\begin{cases}
\dfrac{2^x}{(e^2-1)x!} \quad &\text{if} \, x =1,2,3,\dots \\
0 \quad &\text{if} \, o/w \\
\end{cases}
Find $E(X)$.
$$E(X)=\...
0
votes
2answers
49 views
Write as a power series
I've got to write $\sum_{n=1000}^{\infty} i^{n}\frac{z^{2n-1}}{n!}$ as a power series. However in my mind isn't it already a power series, I don't really understand what the question is asking.
I ...
0
votes
1answer
20 views
Evaluating a derivative in gp/pari
Specifically, I'm trying to evaluate the derivative of the Weierstrass P function at a specific point.
I know that I can set up a function like the following
p(z) = ellwp([1,I],z);
Which will ...
0
votes
1answer
22 views
Radius of Finite Convergence
I am trying to solve a problem where I have to find the radius of finite convergence problem. I believe that I solved the problem correctly, receiving an answer of 1. However, I was informed that this ...
0
votes
1answer
25 views
Writing a complex function as a power series?
I have been asked to write the following summation as a power series:
$$\sum_{n \geq 500} i^n \frac {z^{5n-2}}{n!}. $$
I know that by comparison to the power series $$\sum_{n \geq 0} a_n (z-a)^n, ...
-2
votes
1answer
32 views
For what values of x following function converges absolutely and uniformly? [closed]
$$\sum_{n=1}^\infty\frac{{x}}{1+n^2x}$$
The solution that i obtained was wrong.I thought it will converge for all $x$.
1
vote
0answers
34 views
Show that following series converges uniformly on $[0,1]$ if and only if $α > 1$
$$\sum_{n=1}^\infty\frac{{x^α}}{1+n^2x^2}$$$α>0$
I tried solving it using Weierstrass M test but i didn't got $M_n$ as function of $n$.
0
votes
1answer
29 views
Interchange of a sum and integral
I have to compute the following integral:
$$N = -\frac{2\pi A}{h^2}\int_{0}^{\infty} \ln(1-ze^{-\beta pc})p\mathrm{d}p$$
Let's say $z << 1$. Thus:
$$\ln(1-x) = -\sum_{n=1}^{\infty} \frac{x^n}{...
0
votes
1answer
23 views
Radius of Convergence for two power series is equal.
Given any two power series $\sum_{n=0}^{\infty}{a_n{(z\ -\ a)}^n}$ and $\sum_{n=0}^{\infty}{b_n{(z\ -\ b)}^n}$ , if there is some m ∈ N such that $a_n$= $b_n$ for all n ≥ m, then $\sum_{n=0}^{\infty}{...
0
votes
1answer
48 views
Write … as a power series
i did a workshop recently about writing a series as a power series and then finding the radius of convergence, i'm perfectly happy finding the radius of convergence when it's in power series form $\...
5
votes
4answers
444 views
Why does $\sum_{n\geq0}(1-x)^n=\frac1x$ have such a poor radius of convergence?
I am confused as to why $$\sum_{n\geq0}(1-x)^n=\frac1x$$
only works for $x\in (0,2)$. I get that it has a singularity at $x=0$, so that can't work, but there are no singularities for the rest of the ...
0
votes
1answer
27 views
Using Maple 2018 to find coefficients in a power series
I'm doing a practical experiement and have measured the voltage over a capacitor on an oscilloscope as the capacitor decays.
I have a table of results showing the voltage and time, and created an ...
0
votes
1answer
31 views
Using the Cauchy-Hadamard Theorem to find a radius of convergence
Cauchy-Hadamard Theorem
Consider the formal power series $$f(z) = \sum_{n = 0}^{\infty}
c_{n}(z - a)^{n} $$
for $a, c_{n} \in \mathbb{C}$. Then the radius of convergence of $f$
at the ...
1
vote
3answers
39 views
Radius of convergence of $\sum_0^{\infty} n!x^{n^2}$ [duplicate]
What can be said about the radius of convergence of the poower series
$$\sum_0^{\infty} n!x^{n^2}$$
I know that $\limsup_{n\to\infty}(n!)^{\frac1{n}}\to\infty$. Is that of any use here? Should I use ...
2
votes
1answer
43 views
Interpolation with $p$-adic formal power series in $\mathbb Z_p[[x]]$
It's a classical exercise in complex analysis that one could find a holomorphic function with given values on a set of points on the complex plane without limit points.
What about the p-adic analogue?...
0
votes
1answer
22 views
Radius of convergence for $\ln(a+x)$
Since the radius of convergence $R$ for the Taylor series of $\ln(z)$ around $1$ is $1$, i.e.
$$ R\left\{\ln\left(1+z\right) = \sum \frac{(-1)^{k-1}}{k}x^k \right\} = 1 ,$$
does this mean that for
$...
3
votes
2answers
69 views
Calculate $\sum\limits_{n=0}^{\infty} \frac{x^{3n}}{(3n)!}$ [duplicate]
$$\sum\limits_{n=0}^{\infty} \frac{x^{3n}}{(3n)!}$$ should be calculated using complex numbers I think, the Wolfram answer is :
$ \frac{1}{3} (e^x + 2 e^{-x/2} \cos(\frac{\sqrt{3}x}{2})) $
How to ...
2
votes
0answers
52 views
Do you recognize this infinite series? $\sum_{n=0}^\infty \frac 1{(1+an)^c} \frac{b^n}{n!} $
I came by this infinite series
$$\sum_{n=0}^\infty \frac 1{(1+an)^c} \frac{b^n}{n!} $$
Is there some special function that can have this form?
$c$ can be assumed to be a positive integer. While $...
3
votes
1answer
59 views
Magic relation for harmonic numbers
I was doing same computations for an exercise and I came up with the following relation
$$\sum_{k=1}^n\frac{1}{k}=\sum_{j=1}^n\frac{1}{(j-1)!}\sum_{i_1+\dots+i_j=n\\i_1,\dots i_j\geq1}\frac{1}{i_1i_2\...
