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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Monotonicity of quotient of power series

How can it be proved that $$ f(x) = \frac{\sum_{n=0}^{\infty} \frac{n}{2 x} \frac{(\sqrt{x})^{n}}{\sqrt{n!}}}{\sum_{n=0}^{\infty} \frac{(\sqrt{x})^{n}}{\sqrt{n!}}} $$ is monotonic? The plot of $f(x)$ ...
Guillermo's user avatar
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Convergence of taylor series expansion in the evaluation point

The Taylor series expansion of $f(x)=e^x$ in $a=1$ is: $T(x,a=1)=\sum_{n=0}^{\infty}\frac{e\left(x-1\right)^{\ n}}{n!}$ The interval of convergence using the absolute ratio test is $x\in(-\infty,\...
Freeman's user avatar
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Stanley 's example 1.1.11 and the exponential function

I opened Stanley's Enumerative combinatorics Vol. 1 and looked at example 1.1.11: Suppose $F(0)=1$ and let $G(x)$ be the unique power series satisfying $G'(x)=F'(x)/F(x)$, $G(0)=0$. The author wants ...
undefined's user avatar
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Differentiating a Bessel function power series

For a function defined as $$\begin{align} y(x) &:= \sqrt{\frac{\pi x}{2}} J_{1/2}(x) \\ &= \sqrt{\frac{\pi x}{2}} \sum_{k=0}^{\infty} \frac{(-1)^k}{k! \Gamma (k + 3/2)} \left( \frac{x}{2} \...
spooleey's user avatar
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2 votes
1 answer
73 views

Absolute convergence on boundary implies continuity of power series

Let $f(z) = \sum_{n=0}^{+\infty}c_nz^n$ be a complex power series with radius of convergence $R=1$. Suppose that the series of coefficients converges absolutely, i.e. $\sum_{n=0}^{+\infty}|c_n| < +\...
Matteo Menghini's user avatar
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Is there a closed form for the power series $\sum_{k\geq 1}x^{2^{k}}$? [closed]

I would like to know if anyone could suggest me a reference where the authors discuss and provide a closed form for the power series: \begin{align*} f(x) := \sum_{k\geq 1}x^{2^{k}} \end{align*} where $...
user1234's user avatar
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Summing a nonstandard sequence, closed form of $S_n(x) = \sum_{i=1}^n x^{c^{i-1}}$

Arithmetic sequences have a common difference, where you add a constant to each term to get the next. Geometric sequences have a common ratio, where you multiply a constant to each term to get the ...
Andrew Prudhom's user avatar
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1 answer
81 views

Proof that $e^z=e^xe^{iy}$ [duplicate]

The proof goes like this: $$ \begin{aligned} e^z=e^{x+iy}&=\sum_{n=0}^\infty\dfrac{(x+iy)^n}{n!}\\ &=\sum_{n=0}^\infty\sum_{k=0}^n\dfrac{x^k(iy)^{n-k}}{k!(n-k)!}\\ &=\sum_{n=0}^\infty\...
Joan S. Guillamet F.'s user avatar
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Generalization of binomial coefficients to both non-integer arguments

It is known that binomial coefficients can be generalized to the following: for $s\in\mathbb R$ and $k\in\mathbb N$, \begin{equation*} \binom{s}{k} := \prod_{i=0}^{k-1} \frac{s-i}{k-i} = \frac{s(s-1)...
Dreamer's user avatar
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Behavior of Incomplete Gamma function with negative s, as x goes to zero

From the second item under the Asymptotic Behavior section of the Wikipedia article for the incomplete gamma function, it is written that $$ \frac{\Gamma(s,x)}{x^s} \to -\frac{1}{s} \text{ as } x \to ...
oswinso's user avatar
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Calculate the powers of the sets.

Find the powers of the sets: (a) 𝑋 = {$𝐴 : 𝐴 ⊆ \mathbb R$ and $𝐴$ has a smallest and largest element}; (b) 𝑌 = {$𝐴 : 𝐴 ⊆ \mathbb Z$ and $𝐴$ has a smallest and largest element}; (c) 𝑍 = {$𝐴 : ...
Poetry Lover's user avatar
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Can a nonzero complex power series have an uncountable set of complex roots?

Following Can a real power series have an uncountable number of real roots? and this essentially equivalent question about a sort of linear independence of powers of a real function, the natural thing ...
Olius's user avatar
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In what sense, Taylor series at $x=0$ is better than Taylor series at $x=a$, and vice-versa?

In the book `Calculus Early Transcendentals (6th Edition)' by James Stewart, on p. 739, the author writes: We have two series representations for $e^x$ as follows: $$e^x=\sum_{n=0}^\infty x^n/n! , \...
UAD's user avatar
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Why does this sum not diverge?

So I'm asking this question in general but with a motivating example: Say we have a function $$f(x)=\sum_{n=0}^{\infty}\frac{x^n}{n!}$$ So both the top ($x^n$) and bottom ($n!$) of this function grow -...
Shelby Longbottom's user avatar
1 vote
2 answers
85 views

Computing integral of a series [closed]

Can anyone help solve this problem or provide any advice please? Compute the integral of $f: [0,1] \rightarrow \mathbb{R}$ defined by: $$f(x) = \begin{cases} 0 & \text{if }\; x = 0, \\ ...
qqqqqqqqqq's user avatar
4 votes
2 answers
143 views

Seeking more alternate proofs of a combinatorial generating function identity $G(x)=\overline{G}(-x)^{-1}$ related to counting strings.

Let $\mathcal{S}=[m]^*$ be the set of all strings on the alphabet $[m]=\{1, 2,\cdots, m\}$. Let $\Sigma\subset[m]^2$ be a set of strings of length $2$, and let $\overline{\Sigma}=[m]^2\backslash\Sigma$...
C-RAM's user avatar
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1 vote
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Can $I[[X]] = I\cdot R[[X]]$ if $I$ is not finitely generated?

Let $R$ be a ring and $R[[X]]$ be the formal power series ring. For any ideal $I\subset R$, denote by $I[[X]]$ the ideal of power series with coefficients in $I$. If $I$ is finitely generated, it is ...
Jerry Scott's user avatar
1 vote
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97 views

Why is $\ln2\approx 0.4^{0.4}$?

Just stumbled upon $$\ln2\approx 0.4^{0.4}$$ and wondered if that's just a coincidence, or whether there's some deeper reason? $$\ln2 - 0.4^{0.4}\approx 0.00000234$$ which is a relative error of just $...
emacs drives me nuts's user avatar
2 votes
1 answer
41 views

Solving ODE using power series method but get the general solution $y=0$

Given ODE $$x^2y'-y=0,\qquad x\neq 0.$$ Solve given ODE using power series method. Let $$y=a_0+a_1x+a_2x^2+a_3x^3+\ldots.$$ Then, $$y'=a_1+2a_2x+3a_3x^2+4a_4x^4+\ldots.$$ Now, we substitute to ODE ...
Ongky Denny Wijaya's user avatar
1 vote
1 answer
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Prove that a functional series satisfies a particular functional equation

I have some difficulties with this problem. Let $$g(x)= \lim_{k\to\infty}\sum_{n=-k}^k \frac{1}{x+n}$$ Prove that $$g\left(\frac{x}{2}\right)+g\left(\frac{x+1}{2}\right)=2g(x)$$ I've already proved ...
SeniorWojtasekkk's user avatar
4 votes
1 answer
115 views

How to define $e$ from power series?

I'm trying to take a different approach to show the exponential function and $e$ should exist in calculus. Say, we want there to be a hypothetical function $f$ which should have the property that it ...
rigel's user avatar
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Calculating the sum of $\sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}$ [duplicate]

I want to find the sum of $\sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}$. I have tried to turn it into a power series for a known function, with no luck. I also tried to write it as $\sum_{n=0}^{\infty} \...
Per's user avatar
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why does the formula for the radius of convergence $R^{-1}=\lim_{n\to \infty}\frac{|a_{n+1}|}{|a_n|}$ valid here?

Suppose we have the series of the form $\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1}$. How do I find the radius of convergence? My attempt I used the Ratio test. I got the R=1. But in the textbook ...
Unknown x's user avatar
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A power series solution to wave equation

Find a solution of $$ u_{tt}-u_{xx}=\lambda^2u$$ of a form $$u=f(x^2-t^2)=f(s) \text{ with } f(0)=1,$$ in form of power series in s, where $\lambda$ is a nonzero constant. First thing I do is to find $...
Apple's user avatar
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Domain of convergence for a power series

I have a homework question that we have not discussed at all how to do in class, so any help or explanations would be greatly appreciated! I don't know where to start. In each of the following cases, ...
hii's user avatar
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1 answer
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Series bounded by uniform convergent series is uniform convergent?

I was reading about the Weierstrass $M$-test, and I was wondering about the following: Suppose that $(f_{n})$ is a sequence of real-valued functions defined on a set $A$, and there exists $(g_{n})_{n ...
wsz_fantasy's user avatar
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2 votes
2 answers
93 views

Prove that tan($Z$) = $\frac{e_1t - e_3t^3 + e_5t^5 - \cdots}{1 - e_2t^2 + e_4t^4 - \cdots}$

Let $e_r$ and $p_r$ denote the $r$-th elementary symmetric function and power sum, respectively. Let $t$ be a formal variable and define $$Z := p_1t-p_3t^3/3+ p_5t^5/5- \cdots $$ Prove that the ...
Eduardo4313's user avatar
6 votes
2 answers
186 views

Series with only positive integer coefficients

I want to show that the coefficients of the series $$ f(x) = \sqrt[4]{\frac{1+4x}{1-4x}} = 1 + 2x + 2x^2 + 12x^3 + \cdots $$ are only positive integers. Since differentiating $f(x)$ shows $\left(1 - ...
Dabin Yu's user avatar
3 votes
1 answer
98 views

I can't find the mistake in my proof

I thought about something in power series and I proved a theorem that I have never seen before: The theorem: Let $\begin{align} \sum_{n=1}^{\infty} a_nx^n \end{align}$ be a power series, with a ...
Chess player's user avatar
-1 votes
0 answers
18 views

Are there generalized power-reduction formulae? [duplicate]

I am studying trigonometry and have wondered if there are generalized power-reduction formulae for sin^n(x) and cos^n(x) yet the best that I found was formulae for odd and even powers on Wikipedia ...
Orkeey's user avatar
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2 votes
3 answers
142 views

proof for series to the power of 5 is divisible by n+1 for even n and n congruent to 3 mod 4

I really need help with this: $$\sum_{x=1}^{n} x^5 $$ is divisible by $n+1$ when $n$ is even, and when $ n \equiv 3 \pmod{4} $. What I first did was proving by induction that $$\sum_{x=1}^{n} x^5 = \...
Ari1234567's user avatar
2 votes
2 answers
99 views

Power series of $\frac{1}{7+x}$

For my calculus 2 class, I was given the problem to solve for the power series of $\frac{1}{7+x}$. For my solution, I said $\frac{1}{7+x} = -\frac{1}{1 - (8 + x)}$ and used $8 + x$ as $r$ and $-1$ as $...
ethan warco's user avatar
5 votes
1 answer
106 views

Series expansion of the integral $\int_{-\infty}^{\infty} \frac{\tanh{(x^2-a^2)}}{x^2-a^2} dx$ in the limit of small $a$

I found that when $a \gg 1$, the function $I(a) = \int_{-\infty}^{\infty} \frac{\tanh{(x^2-a^2)}}{x^2-a^2} dx \approx \frac{2\gamma + 2 \log(16/\pi) + 4 \log a}{a}$. How does the integral behave when $...
Archisman Panigrahi's user avatar
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0 answers
96 views

Prove by induction: Let $r \in \mathbb{N}$. $\exists a_{r,1}, \ldots, a_{r,r} \in \mathbb{Q} \, \forall n \in \mathbb{N}$

$$ \sum_{k=1}^{n} k^r = \frac{1}{r+1} n^{r+1} + a_{r,r} n^r + \ldots + a_{r,1} n $$ I'm apparently supposed to use the binomial theorem, but I can't seem to figure it out. The note says: "For the ...
Kadir's user avatar
  • 1
0 votes
2 answers
106 views

Compute the sum of $\sum_{n=1}^\infty \frac{1}{(2n-1)3^n}$ [closed]

I was asked to sum up the series: $$\sum_{n=1}^\infty \frac{1}{(2n-1)3^n}$$ However, I tried to write it: $$\sum_{n=1}^\infty \frac{1}{(2n-1)3^n}=\frac{1}{1*3} + \frac{1}{3*9} + \frac{1}{5*27}+...$$ I ...
Panda Chou's user avatar
4 votes
0 answers
62 views

Apply Power Series Solution Method on Solving ODE with non-integer x order term $ \frac{d^2y}{dx^2}+x^{1/2}y=0 $

Problem: Use series solution expansion to solve ODE in region x>0 $$ \frac{d^2y}{dx^2}+x^{1/2}y=0 $$ Attempts: Assume $y=\sum^\infty_{n=0} a_nx^n$, then $\frac{d^2y}{dx^2}=\sum^\infty_{n=2} n(n-1)...
Yuko's user avatar
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2 votes
1 answer
98 views

Is there such a thing as an intermediate value theorem in complex analysis?

I have an exercise I want to solve, but got stuck in the second part due to not having something like the intermediate value theorem for the complex plane. The exercise is: Let $\Omega$ be an open set ...
Airbornedawn345's user avatar
0 votes
3 answers
201 views

For holomorphic function $f: \Omega \to \mathbb{C}$ with $f^{(k)}(z_0) \in \mathbb{R}$ prove that $f(x) \in \mathbb{R}$ for every $(z_0 - r, z_0 +r)$

I have some difficulties with a question I have come across. The question goes as follows: Let $\Omega$ be an open set with $z_0 \in \Omega \cap \mathbb{R}$. Let $f: \Omega \to \mathbb{C}$ be ...
RIP's user avatar
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2 votes
0 answers
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On the naturality of the definition of a deformation?

Let $(A,\mu_0)$ be an associative unital $k$-algebra over a field $k$. A formal deformation of $(A,\mu_0)$ is sometimes defined as a $k[[t]]$-bilinear map $\mu: A[[t]]\times A[[t]]\rightarrow A[[t]]$ ...
Margaret's user avatar
  • 1,653
1 vote
1 answer
59 views

Operator inverse as a geometric series [duplicate]

Let us consider an operator in the form $$ \left( \mathbf{1} - \lambda \hat{O} \right) \,. $$ Under which circumstances I can write its inverse as $$ \left( \mathbf{1} - \lambda \hat{O} \right)^{-1} = ...
Dario_Maglio's user avatar
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0 answers
60 views

Holomorphic extension of the power series $\sum_{n \geq 0} z^n $

So the power series $\sum\limits_{n \geq 0} z^n $ converges in the unit disc and is holomorphic in the unit disc as well with the derivative $\sum\limits_{n \geq 0} nz^{n-1}$. I am trying to prove ...
mikasa's user avatar
  • 303
4 votes
1 answer
68 views

Example of $a_k$ such that $\sum a_k$ doesn't converge but $\lim_{x \rightarrow 1, x< 1} \sum a_k x^k$ exists.

In class we were taught that if $\sum a_k$ converges, then $\lim_{x \rightarrow 1, x < 1} \sum a_k x^k$ exists. The proof is based on the Cauchy's convergence criterion: if the sum converges, then $...
ABlack's user avatar
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1 vote
0 answers
58 views

How to calculate convergence of a series related to the Euler's constant?

I understand that $$ \sum_{r=1}^n \frac{1}{r} \approx \int_{1}^n \frac{dx}{x} = \log n $$ I would like to verify whether the following holds true as well, for a series that is related to the above ...
Shatarupa18's user avatar
1 vote
1 answer
120 views

Power-series expansion of $\mathrm{e}^{\frac{z}{2}(t-\frac{1}{t})}$

I would like to know the procedure to do the power-series expansion of this function: $$\LARGE{\mathrm{e}^{\frac{z}{2}(t-\frac{1}{t})}}$$ so that I would get this: $$\sum_{r=0}^\infty \sum_{s=0}^\...
user122424's user avatar
  • 3,872
1 vote
0 answers
53 views

How to convert first order singular ODE to integral equation?

I have the following ODE: \begin{equation} v(x)y'=a(x)y^2+b(x)y,~~~y(0)=\varphi,~~v(0)=v(1)=0. \label{eq1} \end{equation} I know that for a similar equation of the form $\varepsilon y'=a(x)y^2+b(x)y~~$...
WhyItDoesNotWorkForMe's user avatar
1 vote
0 answers
70 views

Show that $k[[x,y]]/(y^2-x^2) \simeq k[[x,y]]/(y^2-x^3-x^2)$

I have to show that $$k[[x,y]]/(y^2-x^2) \simeq k[[x,y]]/(y^2-x^3-x^2)$$ where $k[[x,y]]$ is the ring of formal series in $x,y$ over an algebraically closed field $k$. I have the hint that the map $y \...
MathIsNice1729's user avatar
0 votes
0 answers
75 views

$k[[t]]$-module structure on coordinate ring

Let $k$ be a field and $k[[t]]$ the ring of formal power series over $k$. A text that I am reading says: Consider the coordinate ring $k[x,y,t]/(x^2+txy)$ with its canonical $k[[t]]$-module structure....
Margaret's user avatar
  • 1,653
0 votes
0 answers
41 views

Expressing $ \sum_{\lambda\in\text{Spec}(\mathbf{X})} \lambda \log \lambda$

Say $\text{Spec}(\mathbf{X})$ is a set of eigenvalues of a real positive definite matrix $\mathbf{X}$. How do I express $$ \sum_{\lambda\in\text{Spec}(\mathbf{X})} \lambda \log \lambda$$ without ...
AetbeUT's user avatar
  • 525
0 votes
0 answers
63 views

Series of Analytic Functions is Analytic

Let $0 \in \mathbf{N}$. Let $P_m(x): [0,1] \to \mathbf{C}$ be bounded analytic functions for every $m\in \mathbf{N}$. Formally, define $$ f(x) = \sum_{m\in \mathbf{N}}c_m P_m(x)\overline{P_m}(x), $$ ...
Doofenshmert's user avatar
-4 votes
1 answer
163 views

Solution of $ 1 + \frac{x}{2!} + \frac{x^2}{4!} + \frac{x^3}{6!} + \frac{x^4}{8!} + \dots = 0$ [closed]

So according to the book, the solution is true but is this even possible? The equation is: $$1 + \frac{x}{2!} + \frac{x^2}{4!} + \frac{x^3}{6!} + \frac{x^4}{8!} + \dots = 0 \ $$ It asks the solutions, ...
barış yaycı's user avatar

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