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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.

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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
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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?
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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}^{\...
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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 ...
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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 ...
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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 ...
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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
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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 $\...
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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}^{\...
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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 ...
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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
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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 $|x| > 1/\alpha$ ...
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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\...
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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
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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
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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 $$C_{n+1}= -\dfrac{C_n}{2(n+1)}.$$ ...
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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 $...
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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}...
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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.
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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.
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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 ...
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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}{...
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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
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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 ...
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0answers
19 views
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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
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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
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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
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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})...
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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 $a>0$.
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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 ...
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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)=\...
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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 ...
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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 ...
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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 ...
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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, ...
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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 $x$.
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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 $n$.
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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}{...
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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}{...
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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 $\...
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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 ...
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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 ...
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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
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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
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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
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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\...