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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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
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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}...
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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 ...
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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
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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\...
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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 ...
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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 ...
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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^{\...
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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}...
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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 ...
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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 ...
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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 ${𝑓_𝑛}$ converges to 𝑓 uniformly on a ...
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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}...
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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 ...
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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!
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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
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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?
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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 ...
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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 ...
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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)=...
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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
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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 $\left|\frac{a_{n+1}}{a_n}\right|$ ...
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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 ...
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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
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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\...
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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
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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
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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^\...
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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
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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{...
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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, ...
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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.
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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
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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
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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
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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
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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
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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
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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
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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 ...
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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 ...
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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
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0answers
37 views

Efficient way of multiplying two power series expansions? [closed]

I was asked to multiply these two power series expansions: ie. $p(x)q(x)$ $$p(x) = 2(x βˆ’ 4) βˆ’ 7(x βˆ’ 4)^2 + 5(x βˆ’ 4)^3 + . . .$$ $$q(x) = 4 βˆ’ 3(x βˆ’ 4) + (x βˆ’ 4)^3 + . . .$$ and I was wondering if there'...
0
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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
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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 ...
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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} ...
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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 ...

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