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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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23 views

How to find the power series of $\sqrt{1+x^4}$?

The complete question is to find the integral from $0$ to $1$ of $$\sqrt{1+x^4}$$ I am unsure of how to find the power series of this equation in order to do that. I haven't dealt with square root ...
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1answer
39 views

Is there a better way to calculate the Arc-Cosine-Hyperbolic

I am using $ \operatorname{arcosh} x = \operatorname{arcsinh}(\sqrt { x^2-1 } ) $ with bad results. The standard power series $$ \operatorname{arcosh} x = \ln(2x) - \left( \left( \frac {1} {2} \...
1
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1answer
23 views

Finding the first three nonzero terms in the Maclaurin series: $y=\frac{x}{\sin(x)}$

As the title says I would like to find the first three nonzero terms in the Maclaurin series $$y=\frac{x}{\sin(x)}$$ I have the first few terms for the expansion for $\sin(x)=x-\frac{x^3}{6}+\frac{x^...
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1answer
41 views

Series solution about $x=0$ of $xy''-y'+4xy=0$.

I want to find at least one solution of the differential equation $$xy''-y'+4xy=0$$ about the point $x=0$. I identified that $x=0$ is a regular singular point and thus Frobenius Theorem is applicable. ...
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2answers
43 views

Solve functional equation $f(z)=c+zf(z^2)$ with series expansion?

Let the functional equation $(1)$ be given as $$ f(z)=c+zf(z^2) \tag{1}$$ where $c \in\mathbb R$ and $c \neq 0$. How can this functional equation be solved with series expansion (power, Taylor or ...
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1answer
30 views

Is it possible to find a closed form $\sum_{i=0}^n x^{f(i)}$ in general? For $f : i \mapsto i + i^2$?

Let be $f : \mathbb{N} \to \mathbb{N}$, I'm interested if it is possible to find a closed form of $\displaystyle \sum_{n=0}^{p} x^{f(n)}$ for all $x \in \mathbb{C}$ for all $p \in \mathbb{N}$, also ...
2
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2answers
52 views

Power Series expansion of $x\over(1+x-2x^2)$

I am unable to solve this specific problem. The only "notable series expansion" I can use (and know) is $\sum^{+\infty}_0 x^n =$$1\over(1-x)$ I tried several things but none worked. Writing $x\over(...
1
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0answers
66 views

Find the sum of the series $\sum\frac{(-1)^n}{(2n)!+1}$

The whole question looks like- Prove that, $\sum_{n\ge0}\frac{(-1)^n}{(2n)!+1}$ is convergent. Find its value. (Options: $\pi/2$, $4\pi$, $\pi/4$, $2\pi$) I have showed the convergence part. ...
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3answers
46 views

Prove that $\sum_{n=0}^\infty {\frac{x^{2n+1}}{(2n+1)!}} = \frac{e^x - e^{-x}}{2}$

I have been trying to show: $\sum_{n=0}^\infty {\frac{x^{2n+1}}{(2n+1)!}} = \left(\frac{e^x - e^{-x}}{2} \right)$ I have come so far as to show: $\begin{aligned} \sum_{n=0}^{\infty} {\frac{x^{2n+1}}...
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0answers
16 views

Power series solutions for solving ODE

Consider the equation $$ \rho^2t''-2\rho^3t'+[(\epsilon''-1)\rho^2-\frac1{\sqrt{\omega_r}}\rho-l(l+1)]t=0 \tag{1} $$ where $$ t(\rho)=\rho^m\sum_{v=0}^\infty a_v\rho^v \tag{2} $$ and where $m=l+1$ and ...
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0answers
17 views

Power series solution for nonlinear ODE

I am struggling to find a series solution for the initial value problem $y’=2y^2-3y+2$, $y(0)=y_0$. By plugging in a power series, I have simplified and gotten $$\sum_{n\ge0}[(n+1)a_{n+1}-2(a*a)_n+...
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0answers
12 views

Power series as partial sums of coefficients.

Studying real analysis, there's an exercise, number 41 in chapter 10 in Elon Lages Lima's book "Um curso de análise vol. 1", that states: Let $(-r,r)$ be the convergence interval of $\sum a_nx^n$. ...
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0answers
27 views

Why is this a power series?

Finding the power series for: $f(x) = \frac{1}{2 + x^2}$ at $c=0$ Solution after using substitution of a geometric series with $u= \frac{-1}{2}{x^2}$: $$f(x) = \frac{1}{2}\sum_{n=0}^\infty \left(\...
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0answers
12 views

Power Series Solution for deriving a recursion relation

Unfortunately, I'm not familiar with Power series solution method. Also reading some guide files did not help me much. I hope you offer me some hints. I have an equation as follows $$ [- \frac{d^2}{dr^...
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0answers
18 views

How to find: $f^{\alpha}_n(x)=1+\sum_{k=1}^{\infty} \frac{\Gamma(n+\alpha)}{\Gamma(k+\alpha)\Gamma(n-k+\alpha)} x^k$

I am looking for a closed-form solution to $$f^{\alpha}_n(x)=a_0+\sum_{k=1}^{\infty} \frac{a_k\Gamma(n+\alpha)}{\Gamma(k+\alpha)\Gamma(n-k+\alpha)} x^k$$ where we can take $a_k = 1, \forall k\in [0,\...
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0answers
30 views

Summing a general power series

Given a series of the form $$\sum_{r=0}^N f(r)\alpha^r\tag{1}$$ where $f$ is an analytic function of $r$, $N$ is an integer, and $\alpha>0$, how could one solve the above for it's sum? In ...
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3answers
49 views

How can Taylor series diverge?

I understand how to prove a power series diverges, but that seems to contradict simple logic for me. The idea behind constructing a Taylor series is that it is a polynomial that has the same nth ...
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0answers
28 views

Simplifying a marginal likelihood function

I have the following likelihood function $$ p( z \big| x, \lambda, \sigma) = \frac{1}{\sigma^{2}} \cdot \exp \bigg( -\frac{\big( z^{2} + \lambda^{2} \cdot x^{2} \big)}{ \sigma^{2} } \bigg) \cdot I_{0}...
2
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1answer
50 views

How many solutions does equation $\int\limits_x^{x+\frac{1}{2}} \cos \left( \frac{t^2}{3} \right) dt = 0$ have on the segment [0, 3]?

The task i'm trying to solve is: How many solutions (roots) does equation have: $$\int\limits_x^{x+\frac{1}{2}} \cos \left( \frac{t^2}{3} \right) dt = 0$$ on the segment [0, 3] ? By the moment i'...
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3answers
64 views

How do I formally show the radius of convergence of the Taylor series of $f(x)=x^6 - x^4 + 2$ at $a=-2$?

This is an exercise in Stewart's Calculus (Exercise 19, Section 11.10 Taylor and Maclaurin Series): Find the Taylor series for $f(x)$ centered at the given value of a. [Assume that f has a power ...
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2answers
54 views

Can I find a power series representation for $\frac {1}{(1+x)^2}$ ONLY by differentiating $\frac {1}{1+x}$?

There are probably much easier to find the power series for $\frac {1}{(1+x)^2}$ than by differentiating $\frac {1}{1/1+x}$, but I think it should be possible. I've gotten extremely close, but my ...
1
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1answer
33 views

Could the series $\sum (x-3)^n/n$ be seen as a power series if we consider $1/n$ as $c_n$?

I'm just trying to be sure I understand power series correctly. Would the series $\sum \frac{(x-3)^n}{n}$ be seen as a power series if we consider $\frac 1n$ as $c_n$, seeing as (taking $a$ here to be ...
1
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1answer
28 views

Find a composite solution to the following problem

I'm trying to solve the following exercise: Find a composite solution to the following problem: $$ \epsilon y'' + y(y' + 3) = 0 \text{ for }0<x<1, \text{ where }y(0) = 1, \,y(1) = 1 $$ where $\...
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1answer
61 views
+50

Topological Algebraic Independence of power series

Let $p$ be a prime number, let $x$ be a variable, and consider two power series over the ring $\mathbb{Z}_p$ of $p$-adic integers: $a(x):=\underset{n\geq 1}{\sum}{\frac{p^n}{n!}x^n}=px+\frac{p^2}{2}x^...
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2answers
85 views

Derive the sum of $\sum_{i=1}^n ix^{i-1}$

For the series $$1 + 2x + 3x^2 + 4x^3 + 5x^4 + ... + nx^{n-1}+... $$ and $x \ne 1, |x| < 1$. I need to find partial sums and finally, the sum $S_n$ of series. Here is what I've tried: We ...
1
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1answer
41 views

Example III.3.2(1) of Baumslag's “Topics in Combinatorial Group Theory”: proving $F=\operatorname{gp}(1+\xi\mid \xi\in\Xi)$ is free.

This question is a little tricky (for me, at least), since in the textbook the proof of Theorem 5: Every subgroup of a free group is free. is not yet provided (even though I've seen such proofs ...
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2answers
47 views

Complex series involving hyperbolic cosine [on hold]

Please how to calculate the sum of such series! I need the idea ! $$\sum _{n=1}^{\infty} \cosh(n)\frac{z^{2n}}{n!} $$ $$\sum _ {n=0}^{\infty} \frac{(1+i)^{n}z^{n}}{n!}$$
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1answer
30 views

Find the Taylor series of $\frac{1+z}{1-z}$ at $z_{0}=i$

I'm given the following explanation: Let $\dfrac{1+z}{1-z} = \biggl(\dfrac{1+i}{1-i}+\dfrac{z-i}{1-i}\biggr)\biggl(1-\dfrac{z-1}{1-i}\biggr)^{-1}$ = = $\dfrac{1+i}{1-i}\displaystyle\sum_{j=0}^{\...
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1answer
26 views

What is the radius of convergence for $\sum_{j=0}^{\infty}\frac{i^{j}+(-i)^{j}-i}{j!}z^{j}?$

How do I find the radius of convergence for \begin{equation} \sum_{j=0}^{\infty}\frac{i^{j}+(-i)^{j}-i}{j!}z^{j} \end{equation} At first I thought I could do some simplification using the fact that ...
2
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3answers
417 views

How does this infinite series $1-\frac{1}{4}+\frac{1}{7}-\frac{1}{10}+\cdots$ simplify to an integral $\int_0^1\frac{dx}{1+x^3}$?

How does the infinite series below simplify to that integral? $$1-\frac{1}{4}+\frac{1}{7}-\frac{1}{10}+\cdots=\int_0^1\frac{dx}{1+x^3}$$ I thought of simplifying the series to the sum to infinity of ...
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2answers
20 views

Expression of the sum of a power series

I was working on a mathematical stats problem and I don't get this part (that comes from a recursive arithmetico-geometric series): $$U_{k} = \alpha C^{2} + \alpha^{2} C^{2} + \alpha^{3} C^{2} + ... +...
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1answer
24 views

Question regarding complex power series

Let us define a function $f$ on unit disc by- $f(z)=\sum z^{2^n}\ \forall |z|<1$ Let $\theta=2\pi p/2^k$ where $p,k$ are positive integers and $0\le r<1$ so $f(re^{i\theta})$ exists. Show that, $...
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2answers
63 views

Apparent paradox with power series convergence

I'm having problem figuring out how these three facts are mutually consistent: Call a set $S \subset \mathbb{C}$ to be concirclic if for some real number $R$, if you let $A$ be the (open) ball ...
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3answers
52 views

Is The Series $\sum_{n=1}^{\infty} \frac{n^4-3n+2}{4n^5+7}$ Divergent

This is my solution for $\sum_{n=1}^{\infty} \frac{n^4-3n+2}{4n^5+7}$. First I let the sequence in the series be labeled $A$, then I constructed a new sequence ($B$) that would have the similar ...
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3answers
41 views

Evaluate $\sum^{20}_{r=0} \binom{20}{r}\cos r \theta$

Evaluate $$\sum^{20}_{r=0} \binom{20}{r}\cos r \theta$$ To do this my initial thought was to set up an alternative series, $S$, $$S=\sum^{20}_{r=0} \binom{20}{r}\sin r \theta$$ If yoy let the ...
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1answer
26 views

Power series convergence (center of the power series)

Does every power series converge at at least one point? Would this point be the point at which the power series is centered. Also, is there a proof for the above question?
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1answer
15 views

Power series modification - two methods, different results?

First, there was a question that stated: $g(x) = \frac{1}{1+x}$, with the power series representation being $\sum_{n=0}^\infty (-1)^nx^n$. The second question was as follows: find the power series ...
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1answer
43 views

Power Series Representation of $\frac{1}{(1+3x)^2}$

Question: Find a power series representation of the function: $$ f(x) = \frac{1}{(1+3x)^2}$$ through the use of an anti-derivative. My Approach: Take the integral of f(x), Result ==> $\frac{-1}{3}\...
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0answers
15 views

Series solution of first order differential equation with constant term

I am curious about whether one can find a series solution to the differential equation $$y’=f(y,t)+k \ \ \ \ k \in \mathbb{R}$$ I know the standard way of solving a differential ...
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0answers
34 views

Finding a second solution for a differential equation with regular singular point

The differential equation: $xy'' + y =0$ has a regular singular point at $x=0$, using Frobenius' method, we get only one series solution which is: $$ y_{1} (x) = x - \frac{1}{2} x^2 + \frac{1}{12} x^...
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2answers
82 views

What is the generating function of $z^3 + 2z^5 + 5z^7 + 14 z^9+\dots$?

What is the generating function of $z^3 + 2z^5 + 5z^7 + 14 z^9+\dots$ ? The generating function can be written as follows: $$A(z)=\sum_{i>2}^{\infty} a_i z^{2i+1},\text{where } a_i \text{ is the ...
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2answers
60 views

Bessel function identity

I was trying to find this identity of Bessel function $$e^{-2i\gamma t} J_{\left|n\right|}(2\gamma t) = e^{\large \frac{\pi i}{2}} \sum_{k=|n|}^{\infty} \frac{(-i\gamma t)^k}{k!}\binom{2k}{k-n}$$ on ...
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1answer
30 views

Expand $f(z)=\frac{1}{z(z-3)}$ as laurent series in domain $1 < |z-4| < 4$

Expand $f(z)=\frac{1}{z(z-3)}$ as laurent series in domain $1 < |z-4| < 4$ Any suggestion i have $\frac{1}{z-3}= \sum_{n=0}^{\infty} (-1)^n \frac{1}{(z-4)^{n+1}}$ and $\frac{4}{z}= \...
6
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2answers
129 views

Proving that $\sum_{n=1}^\infty \frac{\sin^2 n}{n^2}=\sum_{n=1}^\infty \frac{\sin n}{n}$.

Proving that $$\sum_{n=1}^\infty \frac{\sin^2 n}{n^2}=\frac{\pi -1}{2}$$ I've known a similar conclusion $$ \sum_{n=1}^\infty \frac{\sin nx}{n}= \begin{cases} \dfrac{\pi - x}{2} & x \in (0, 2\pi),...
0
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1answer
44 views

Finding a power series using differentiation

This question has been asked already on this site, but I found the solution confusing relative to my understanding. Use differentiation to find a power series representation for: $f(x)=\frac{1}{(8+x)^...
1
vote
1answer
52 views

$f(x) = \frac{x}{2x^2+1}$ Power Series Interval of Convergence

Alright, so I have a homework problem and I'm struggling on finding the interval of convergence/just need some reinforcement on this concept to garner a stronger grasp on how to solve these problems. ...
0
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1answer
31 views

Expand $\frac{1}{z^2}$ as a series on $|z - 2| < 2$

I am asked to show that $$\frac{1}{z^2} = \frac{1}{4} + \frac{1}{4}\sum_{n = 1}^{\infty}(-1)^{n+1}(n+1)\left(\frac{z-2}{2}\right)^n$$ on $|z - 2| < 2$. My plan is to consider a series expansion ...
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2answers
90 views

Representing power series as a function - what to do with the constant after integration?

This power series $$f(x)=\sum_{n=1}^{\infty} {\frac{x^{3n}}{3n}}$$ when differentiated, loses $3n$ in the denominator, with one manipulation, one can get $$f'(x)=\frac{1}{x(1-x^3)} $$ using the ...
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0answers
5 views

Perturbative expansion of exponential integrals

Suppose that I have a sigma-finite measure space $(X,\Sigma,\mu)$ and a fuction $P:X\rightarrow\mathbb{R}$. Then, is it true that $$\int\text{d}\mu(x)e^{P(x)}=\sum_{k=0}^\infty\frac{1}{k!}\int\text{d}\...
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0answers
18 views

Chain Rule for $\mathbb Q[[T]]$

I am trying to prove the chain rule for formal power series $\mathbb Q[[T]]$, i.e.: Given $f(T) = \sum_{n \geq 0}f_nT^n, \; h(T) = \sum_{n\geq0}h_nT^n$, then: $\frac{d}{dT} f(h(T)) = \left(\frac{d}{...