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.
6,572
questions
2
votes
2answers
41 views
Proving $f(x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + … $ is continuous for a fixed $x_0 \in (-1,1)$ by using the Weierstrass M-test
I am trying to prove that $f(x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + ...$ is continuous for a fixed $x_0 \in (-1,1)$ by using the Weierstrass M-test. Now, the solution in the book is ...
-1
votes
1answer
20 views
Prove: $\sum_{n=0}^\infty \frac{a_n}{n!}x^n$ converges for every $x$ if $\sum_{n=0}^\infty a_nx^n$ has radius of convergence $R>0$
$\sum_{n=0}^\infty a_nx^n$ has radius of convergence $R>0$
Prove: $\sum_{n=0}^\infty \frac{a_n}{n!}x^n$ converges for every $x$.
Also, I don't understand why $R < 1$ implies $$\lim_{n \to \infty}...
0
votes
3answers
22 views
Analytic form of power series model?
The Setup
I am dealing with the recursive equation
$$w_t = \alpha + (1-\alpha)w_{t-1}, $$
with $0 < \alpha < 1$ and some starting point $w_0$. I am trying to prove an expression for $w_t$ that ...
0
votes
0answers
24 views
Taylor series coefficient calculation
Let $f,\phi$ be two functions holomorphic on the open simply connected $\Omega$ with $0\in\Omega, \phi$ not identicaly $0$ with $f(z)=\phi(z)h(z)$, $h$ holomorphic in $\Omega$ and $g(z)=\frac{1}{z}\...
2
votes
2answers
45 views
Prove that a power series and its deriviative has the same convergence radius
Let $ \sum_{n=0}^{\infty}a_{n}x^{n} $ be a power series. prove that
$ \sum_{n=0}^{\infty}a_{n}x^{n},\sum_{n=0}^{\infty}\left(n+1\right)a_{n+1}x^{n} $
has the same convergence radius.
So, let $ \limsup\...
0
votes
0answers
11 views
Algebraicity of a double series
Let $k=\mathbb F_q(T)$. Is the series $\alpha(X,Y):=\sum_{n\ge0}\frac{Y^{q^n}}{1-TX^{q^n}}$ algebraic over $k(X,Y)$? I think no.
Here it is what I did. Is it correct?
If $\alpha(x,y)$ is algebraic ...
1
vote
1answer
47 views
Prove: if $\sum^\infty_{n=0}a_nx^n$ converges for every $x$, then $\sum^\infty_{n=0}a_n$ converges absolutely
Prove:
if $\sum^\infty_{n=0}a_nx^n$ converges for every $x$, then $\sum^\infty_{n=0}a_n$ converges absolutely.
I get why the statement is correct (because it means that the convergence of the series ...
0
votes
0answers
53 views
Stuck with integrals
I would like to find a "simple" series form for the following integrals:
\begin{align}
I_1(x)&=\int_{t=0}^{\frac{\pi}{2}} t e^{-ix\cos{t}}\,dt\\
I_2(x)&=\int_{\phi=0}^{2\pi} \int_0^{\...
-2
votes
0answers
19 views
Let $a_n$ be a sequence of real numbers such that $\displaystyle\lim_{n\to\infty}a_nx^n=0$ for all $x>0$ [closed]
Let $a_n$ be a sequence of real numbers such that $\displaystyle\lim_{n\to\infty}a_nx^n=0$ for all $x>0$. Also if $R$ is the radius of convergence of the power series $\displaystyle\sum\limits_{n=1}...
1
vote
2answers
87 views
Evaluating $ \lim_{x \to 0}\left(-\frac{1}{3 !}+\frac{x^{2}}{5 !}-\frac{x^{4}}{7 !}+\frac{x^{6}}{9!}+\cdots\right) $
This question comes to my mind immediately after asking this question.
I was earlier unknown that limit of sum equal sum of limits only when there are finite terms. Now the problem is then how do I ...
2
votes
2answers
83 views
If we pick a sequence of numbers $(a_k)$ at random, what is the expected radius of convergence of $\sum_k a_k x^k$?
Suppose we pick a sequence of positive integers independently and identically distributed from $\mathbb{N}^+$: call it $(a_k)=(a_0,a_1,a_2,a_3,\ldots)$. If we consider the corresponding generating ...
1
vote
3answers
58 views
Show that the power series $\sum_{n=0}^\infty \frac{z^n}{n!}$ converges uniformly for all $z$.
Show that the power series $\sum_{n=0}^\infty \frac{z^n}{n!}$ converges uniformly for all $z$.
I know that the definition for uniform convergence is: We say ${š_š}$ converges to š uniformly on a ...
0
votes
2answers
61 views
Generalisation of $\log(1-x)$ expansion?
The standard logarithm expansion is given by
\begin{equation}
\log(1-x) = -\sum_{n=1}^{\infty}\frac{x^n}{n}, \qquad |x|<1
\end{equation}
I have a sum which is of the form
\begin{equation}
\sum_{n=1}...
10
votes
2answers
155 views
Suppose that $f$ is an entire function satisfying $f(2z)=\frac{f(z)+f(z+1)}{2}$. Show that $f$ is constant. [duplicate]
I've been working old qualifying exam problems from my university. I've been struggling with the following:
Suppose $f$ is an entire function with the property that $f(2z)=\frac{f(z)+f(z+1)}{2}$ for ...
-3
votes
1answer
29 views
Find the closed form of $G ( t ) = \sum _ { n = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { n } ( n + 2 ) } { ( 2 n ) ! } t ^ { n }$ [closed]
$$G(t) = \sum_{n = 0}^{+ \infty} \frac {(- 1)^{n} (n + 2)} {(2 n)!} t^{n}$$
Please help me to find the closed form. Thank in advance!
0
votes
1answer
33 views
Find a closed form of this power series f(x). [closed]
$$f ( x ) = \sum _ { n = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { n } ( 2 n + 3 ) } { n ! } x ^ { 2 n }$$
Please help to tell me the methods to find a closed form of the infinite series.
Thank ...
2
votes
0answers
34 views
Write this integral as a power series
I have the integral $$\int_h^{h+\epsilon\delta h}F\,{\rm d}z,$$ which I would like to represent as a power series of $\epsilon$. How might I do this?
0
votes
1answer
29 views
Limits:my question is why can't the numerator be equal to $1/4$ as it forms Grandi Series?
$\lim_{n \to \infty} \frac{(1-2+3-4+5-...-2n)}{(1+n^2)^.5}$
can be solved as $\lim_{n\to\infty}\frac{-n}{(1+n^2)^.5}$
$\lim_{n\to\infty} \frac{-n}{n(1/n^2+1)^.5}$
$=-1$
my question is why can't the ...
0
votes
3answers
53 views
How to find a power series for a given function?
For homework I was asked to derive a power series for the function $e^{2x}-\frac{1}{x}$ centered at x=1 and I'm honestly clueless as to how to do it. I tried writing out the derivatives and I couldn't ...
0
votes
1answer
24 views
How to prove the Taylor series converges for a smooth function with nonnegative derivatives? [duplicate]
Suppose $f\in C^{\infty}[a,b]$, and for any $n\geq 1$, $f^{(n)}(x)$ is non-negative. Prove that
$$f(x)=\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x-a)^n\,,\, \forall x\in [a,b]\,. $$
Hint: using $R_n(x)=...
5
votes
1answer
62 views
Finding the Laurent Series of $f(z)=\displaystyle\frac{z^2 e^{1/z}}{z-1}$ for $0<|z|<1$
I have to calculate the Laurent series at the origin of $f(z)=\displaystyle\frac{z^2 e^{1/z}}{z-1}$ for $0<|z|<1$, my idea was to use the Cauchy product, but I don't know if it is correct.
$$\...
4
votes
2answers
64 views
Is the radius of convergence related to the ratio limit or half of the interval of convergence?
I have a series $S$ with general terms $a_n=\frac{(-1)^n(x-1)^n}{(2n-1)2^n}$, $n\ge 1$:
$$S = \sum_{n=1}^\infty \frac{(-1)^n(x-1)^n}{(2n-1)2^n}$$
Finding the ratio $\left|\frac{a_{n+1}}{a_n}\right|$ ...
2
votes
2answers
39 views
How to write this series as a ratio of polynomials?
I have the series:
$g(z) = \sum^{\infty}_{0} b_{n}z^{n}$ where $z\in$$\mathbb{C}$.
$b_{n}$ is the $n^{th}$ number in the sequence, i.e. $(b_{0},b_{1},b_{2},b_{3},...)$ = (1,1,2,3,...)
How can I write ...
0
votes
0answers
26 views
Finding the Accuracy for a Taylor Polynomial
Let
$f(x)= x^{\frac 1 3}$ at $a=3$ with $n=4$
At the fourth degree ($n=4$) I got the polynomial
$$T_4(x)=3^{\frac 1 3}+\frac{3^{\frac 1 3}}{9}(x-3)-\frac{3^{\frac 1 3}}{81}(x-3)^2+\frac{5\times 3^{\...
4
votes
1answer
150 views
Can this integral be expanded as a power series?
I am looking at this integral:
$$I(x_1^2, x_2^2) = \frac{1}{4}x_1^2 x_2^2 \int\limits_{-\infty}^{\infty} d\tau_3 \int\limits_{-\infty}^{\tau_3} d\tau_4 \int\limits_{-\infty}^{\tau_4} d\tau_5 \int\...
0
votes
4answers
55 views
Find the Sum of the Series $\sum_{n=0}^\infty \frac{3n^2 -1}{(n+1)!}$
Find the Sum of the Series $$\sum_{n=0}^\infty \frac{3n^2 -1}{(n+1)!}$$ I separated the Series in to the sum of $\sum_{n=0}^\infty \frac{3n^2}{(n+1)!}$ and $\sum_{n=0}^\infty \frac{-1}{(n+1)!}$. First ...
2
votes
1answer
55 views
Convergence of the series $\sum_{n=1}^k\frac{k(n+1)}{k-n+1}\theta^{n}$
I am trying to show that this inequality holds
$$
\sum_{n=1}^k\frac{k(n+1)}{k-n+1}\theta^{n}<\frac{c}{(1-\theta)^2},\forall k>0,\theta\in(0,1)
$$
where $c$ is some constant. I have done some ...
2
votes
1answer
47 views
What is wrong in the following method of obtaining the Maclaurin series of $\frac{2x}{e^{2x}-1}$?
$\frac{2x}{e^{2x}-1} = -2x(1-e^{2x})^{-1}$
We can obtain the binomial series expansion of $(1-e^{2x})^{-1}$:
$(1-e^{2x})^{-1} = \sum_0^\infty\begin{pmatrix}-1\\n\end{pmatrix}(-e^{2x})^{n} = \sum_0^\...
0
votes
1answer
224 views
How can I prove that this function is bounded?
The binomial series of $(1-x^n)^{\frac{1}{n}}$, where $n$ is a positive integer, converges absolutely to $(1-x^n)^{\frac{1}{n}}$ for $x\in[0,1]$. The binomial series expansion is
$$(1-x^n)^{\frac{1}{n}...
3
votes
1answer
37 views
Show that if $|z| < 1$ then the series $\sum_{n=0}^{\infty}(n+1)z^{n}$ converges, and find its sum.
Show that if $|z| < 1$ then the series $\sum_{n=0}^{\infty}(n+1)z^{n}$ converges, and find its sum.
MY ATTEMPT
The given series indeed converges. This is a consequence of the ratio test:
\begin{...
0
votes
3answers
38 views
Is there an efficient way to calculate the following power series?
I want to find coefficients of a power series $K_p$ given by the equation:
$$\frac{1-z^2}{1+z^2-2z\cos(\theta)} = \sum_{p=0}^{\infty}K_pz^p$$
where $\theta$ is a constant. I have checked that $K_0=1, ...
1
vote
1answer
40 views
Polynomial bound for complex exponential function
Can you help me prove that $\vert \exp(it) - 1 -it \vert \le t^2$, where $t > 0$? Thank you in advance.
1
vote
1answer
23 views
Calculate a meromorphic function on $\mathbb{C}$ s.t. $g(z) = \sum_{n\geq0}(n+1)T^n$
I have the following power series $f(T)=\sum_{n\geq0}(n+1)T^n$ and I have to calculate a meromorphic function $g$ on $\mathbb{C}$ such that $\forall z \in B_1(0), g(z)=f(z)$
I have tryed the following:...
1
vote
1answer
17 views
Root test for complex series and cancelling powers with absolute values
The root test for convergence of a complex power series is given as
$$\lim_{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|} = L$$
If $a_n = \frac1{(1+i)^n}$ then I read that when applying the root ...
2
votes
1answer
53 views
Find multiplicative inverse of formal power series
I have the following power series:
$f(T) = \sum_{n\geq 0}(n+1)T^n \in \mathbb{C} [T]$
and I have to compute the multiplicate inverse.
I've tryed to use the formula to calculate the coefficients of the ...
0
votes
1answer
71 views
Finding the power series solution to ${(x^2+1)y'' + xy' + 2xy = 0}$ [closed]
Find the power series solution to the following initial value problem about the point ${x=0}$:
$${(x^2+1)y'' + xy' + 2xy = 0}$$
subject to
$${y(0) = 2}$$
$${y'(0) = 3}$$
My attempt:
I tried to find ...
0
votes
1answer
52 views
Show that $\sum_{n = 2}^{\infty} \left(\frac{(-1)^{n}}{(n-1)n} - \frac{9(-1)^n}{2\cdot 3^n}\right)x^n > 0$ for positive $x$.
How can one prove that $$\sum_{n = 2}^{\infty} \left(\frac{(-1)^{n}}{(n-1)n} - \frac{9(-1)^n}{2\cdot 3^n}\right)x^n > 0\,?$$
this inequality resulted after expanding the difference of two functions ...
0
votes
2answers
43 views
$\sum_{n=1}^{\infty}(a_{n}-a_{n-1})x^n$ and $\sum_{n=0}^{\infty}a_nx^n$ has the same convergence radius [closed]
Proof:
Suppose power series $\sum_{n=0}^{\infty}a_n x^n $, if series ${a_n}$ is divergent. Then $\sum_{n=1}^{\infty}(a_{n}-a_{n-1})x^n$ and $\sum_{n=0}^{\infty}a_nx^n$ has the same convergence radius.
...
0
votes
1answer
18 views
Power series problems
How can I find the power series expansion of following functions. I don't have any idea, these seem intimidating. Please help how to proceed.
$(x+\sqrt{1+x^2})^a$
$\sqrt{\frac{1-\sqrt{1-x}}{x}}$
0
votes
1answer
21 views
Power Series ODE Question - Final Step
I just started learning how to use the power series method to solve ordinary differential equations and this is one of the first questions we were asked. I've managed to get it right up to the last ...
0
votes
0answers
11 views
Behavior of the wavefunction at points where potential is singular
I am a physics student trying to understand the behavior of a wavefunction $\psi(x)$ at points where the potential $V(x)$ is singular.
In Griffith's Inroductuion to Quantum Mechanics book, he wrote
...
0
votes
4answers
57 views
Can I find a power series expansion of this function without a Taylor series?
I was asked to find the power series expansion of $f(x) = \frac{x}{\sqrt{x^2+4}}$ about $x = 0$. Is there a way to do a power expansion without finding the Taylor series? Deriving this function ...
2
votes
0answers
37 views
Efficient way of multiplying two power series expansions? [closed]
I was asked to multiply these two power series expansions: ie. $p(x)q(x)$
$$p(x) = 2(x ā 4) ā 7(x ā 4)^2 + 5(x ā 4)^3 + . . .$$
$$q(x) = 4 ā 3(x ā 4) + (x ā 4)^3 + . . .$$
and I was wondering if there'...
0
votes
1answer
22 views
Proof, that multiplying a power series with $\frac{1}{x^k} doesn't change its radius of convergence.
Let $f: x\mapsto \sum_{i = 0}^{\infty} a_nx^n$ be a power series with convergence radius $R_f$ and $k\in\mathbb{N}$, where $a_i = 0,\ \forall i=0,...,k-1$.
Multiplying the series with $\frac{1}{x^k}$ ...
0
votes
1answer
31 views
Inequality from Geometric Series
My Calculus II textbook shows the following example:
I do not understand how they got -2 from solving the inequality $\left|\left(\frac{x}{2}\right)^2\right| < 1$. Doesn't it result in an imaginary ...
0
votes
1answer
27 views
Convergence Theorem for Power series : radius of convergence and normal convergence
I am currently working through the textbook "Complex Analysis" by Freitag and Busam.
Proposition III.2.1 (Convergence Theorem for Power Series) reads :
For each power series
$$ a_{0} + a_{1} ...
0
votes
0answers
32 views
Is there a thorough list of power series along with their corresponding formulas?
About half an hour ago I made a post asking for a proof of the formula $$\sum_{k=0}^{\infty}F_{mk}z^k=\frac{F_mz}{1-z(F_{m-1}+F_{m+1})+(-1)^mz^2}$$
where $F_i$ denotes the $i$-th Fibonacci number, and ...
0
votes
1answer
35 views
Reproducing a function as a Fermi distribution
I have a function such as
$$ g(\epsilon) = \frac{1}{2}\left[ 1 - \frac{\epsilon - \mu}{\sqrt{(\epsilon - \mu)^2 + \Delta^2}}\right]. $$
When $\Delta \rightarrow 0$, $g(\epsilon)$ reproduces very well ...
1
vote
1answer
34 views
Proving $\sum_{k=0}^{\infty} F_{mk}z^k=\frac{F_mz}{1-z(F_{m-1}+F_{m+1})+(-1)^mz^2}$
I have read in a few places that$$\sum_{k=0}^{\infty} F_{mk}z^k=\frac{F_mz}{1-z(F_{m-1}+F_{m+1})+(-1)^mz^2}$$where $F_i$ denotes the $i$-th Fibonacci number. The series is a generalization of the more ...
0
votes
1answer
25 views
If $f:\mathbb R^2 \to \mathbb C$ is real analytic at point implies its real part function also?
A complex valued function $f,$ defined on an open set $E$ in the plane $\mathbb R^{2}$, is said to be real analytic on $E$, if to every point $(s_{0}, t_{0}) \in E,$ there corresponds an expansion ...
