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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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24 views

$t$-adic algebras

At the moment I am studying on $t$-complete $k[[t]] $-algebras and I have asked myself the following question: If $k$ is a field of characteristic zero and $S$ a flat, $t$-complete $k[[t]]$ Algebra, ...
4
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1answer
52 views

A Tauberian theorem for a quotient of power series, the limit on the boundary

Take sequences $a_n, b_n \in \mathbb R_{>0}$ (or $\mathbb C$) such that the limit $$L = \lim_{N \to \infty}\frac{\sum_{n \leq N} a_n }{\sum_{n \leq N} b_n}$$ exists and such that the power series ...
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1answer
43 views

What is a binary expansion of a real number?

Related to this question I asked. I want to know what is exactly meant by a binary expansion of a number, real or natural. Can someone show me via example, how would you binary expand a real number ...
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1answer
42 views

Integrating the geometric series

We Know ($ z \in \textbf{C} $): $\dfrac{1}{1-z} = \sum_{n=0}^{k} z^{n} + \dfrac{z^{k+1}}{1-z}$ Integrating this along the straight line $L$ from $0$ to $z$: $ -ln(1-z) = \sum_{n=0}^{k}\dfrac{z^{n+1}...
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1answer
40 views

Use Differentiability of Power Series to find the sum $F(x) = \sum_{n=0}^{\infty} \frac{(x+1)^{n+1}}{n+1}$

$$F(x) = \sum_{n=0}^{\infty} \frac{(x+1)^{n+1}}{n+1}$$ I found the following values after differentiating: $$ (x+1)^n, n(x+1)^{n-1}, n(n-1)(x+1)^{n-2} $$ It looks a lot like the Sum for $F(x)= 1/(...
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0answers
58 views

Convergence of the Power Series for $xu''+\sin(x)u=0$

Consider the initial value problem $$xu''+\sin(x)u=0 \ \ \ \ u(0)=0, u'(0)=2$$ What can be said about the radius of convergence of this series? I have determined that the first four nonzero terms ...
2
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1answer
58 views

Power Series Solution [duplicate]

Consider the initial value problem $$xu''+\sin(x)u=0 \ \ \ \ u(0)=0, u'(0)=2$$ Derive the first $4$ non-zero terms of a power series solution to this problem about the point $x=0$. I know the ...
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2answers
45 views

How can we prove that $z^{n+1} \to 0_{\mathbb{C}}$ as $n\to \infty$?

In the book of Function of One Complex Variable by Conway, at page 31, it is given that However, normally, if $z$ was a real number, we could argue that $z^{n+1}$ goes to zero as $n \to \infty$ if $|...
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5answers
2k views

Does this mathematical technique have a formal name? And why does it work?

When I split a number in the powers of 2. I am able to make any combination of any number that is less than it by taking each number of the series only once. For example: $7=1+2+4$ I can construct ...
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4answers
31 views

Power Series Expansion in Inverse Powers

In a book I am reading at the moment it says that the expression $$ H = \sqrt{P^2 c^2 + m^2 c^4} $$ can be "formally expanded in inverse powers of $c$" to obtain $$ H = mc^2 + \frac{1}{2m} P^2 + \...
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1answer
38 views

What is the interval of convergence for power series

I have a power series and I need to find the interval of convergence. $$\sum_{n=0}^\infty= \frac {n}{(n^2-1) (1+x)^n} $$ I tried ratio test with a new variable $ t = \frac{1}{1+x} $and I get that ...
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1answer
29 views

Is $\limsup |C_n|^{\frac{1}{n}} = \limsup |C_{n+j}|^{\frac{1}{n}}? $ where $j\geq0 $

$C_n$ is a sequence of complex numbers. I would like to prove that the radius of convergence of a differentiated power series is the same as the original series. i.e. $\limsup |(n+j)(\cdots)(n+1)C_{...
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1answer
36 views

Can someone explain graphically Taylor Remainder Theorem?

Taylor Remainder Theorem is $$|R_n| = \frac{M}{n+1}|x-a|^{n+1}$$ Well, can someone explain this to me more graphically? If $M$ is some number slightly bigger than the $(n+1)^{th}$ order of ...
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1answer
23 views

Sequence bounded and series diverges implies $R=1$.

There is an exercise in Lay's Real Analysis (8.3.7) which runs: Suppose that a sequence $(a_n)$ is bounded but that the series $\sum a_n$ diverges. Prove that the radius of convergence of the power ...
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0answers
65 views

Closed form of $\sum_{n = 1}^N \frac{x^{n-1}}{n!}$ [duplicate]

Is it possible to find a closed form for the following "finite sum"? $$ \sum_{n = 1}^N \frac{x^{n-1}}{n!}. $$ What I came up to: $$ \sum_{n = 1}^N \frac{x^{n-1}}{n!} = \frac{1}{x}\left( \sum_{n = 1}^...
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2answers
30 views

How to multiply in the formal Laurent series ring

I'm working on a problem that asks me to show that the ring of formal Laurent series $F((x))$ is actually a field when $F$ is a field. My problem is that the problem doesn't define the product between ...
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0answers
25 views

Expansion of finite and infinite summations raised to a power

I need to simplify the expression written below to get the $ x $ term in its simplest form: $$ \ \left(\sum_{k=1}^b \sum_{n=0}^\infty C_n(a,k) x^\frac{n+k}{2} \right)^t ,\ $$ where $$ \ C_n(a,k)=\...
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1answer
37 views

Log utility function and the St. Petersburg paradox

In the log utility model the formula solving the St. Petersburg paradox $$\Delta E(U)=\sum_{k=1}^\infty \frac{1}{2^k}\left [\ln(w + 2^k - c) - \ln(w) \right ]$$ relates the wealth, $w,$ of the ...
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0answers
35 views

$L^2$-convergence versus locally uniform convergence for a complex power series

Let $M$ be a nice topological space with a nice measure, let's say a Riemannian manifold. Let $B(0,R)$ be a ball in $\mathbb C$ and suppose $a_n : M \to \mathbb C$ are continuous $L^2$ functions such ...
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2answers
57 views

Laurent series expansion (example)

$$ \text{Let} \quad f(z)=\frac{1}{z}+\frac{2}{3-z},\quad z\in \mathbb{C}\smallsetminus \{0,3\}.$$ To find the Laurent series expansion of $f$ about $z_0=1$ within the annulus $1<|z-1|<2,$ one ...
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1answer
56 views

Evaluating Sum of $\dfrac{i}{(-x)^i}$ [duplicate]

I would like to ask if the expression below can be simplified using standard summation properties? Or should I dive into much deeper concepts like the power series? Thank you. $$\sum_{i=1}^{n-1} \...
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0answers
35 views

Is there accepted notation for the various “duals” of a polynomial? Does any interesting theory surround them?

Let $k$ denote a base field. There's a variety of interesting ways to turn polynomials in $k[x]$ into a "dual operator" that acts maps $k$-linearly from $k[[x]]$ to another $k$-module, such as $k[[x]]$...
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1answer
25 views

What does in Taylor series, what does “Taylor polynomial of f at a” actually mean?

For example, I thought Taylor polynomial of cosine centered at $\frac{\pi}{2}$ meant $\cos (x-\frac{\pi}{2})$. But when expanded with $(x-\frac{\pi}{2})^n$, of which $T_3$ becomes $-(x-\frac{\pi}{2})...
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0answers
16 views

Taylor polynomial remainder theorem, why is $R_n^{n+1}(x) = f^{n+1}(x)$?

I am trying to grasp the concept of Taylor remainder. Online lecture taught me that in equation $R_n^{(n+1)}(x) = f^{(n+1)}(x) + T_n^{(n+1)}(x)$, $T_n^{(n+1)}(x)$ becomes zero because in general ...
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1answer
49 views

Rearranging a power series centered in $z_0$ to a power series centered in 0: is that possible?

Suppose we have this power series $S(z) = \sum_{n=0}^{\infty}a_n(z-z_0)^n$ that converges $\forall z \in D(z_0,r)$, $r$ being the radius of convergence. Is that possible to transform $S(z)$ to $T(z) = ...
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1answer
33 views

If a sum of $L$ positive integers grows like $L^d$, how does the summand grow?

Suppose that $(a_N)_{N \in \mathbb{N}}$ is a sequence of (strictly) positive integers which satisfies the following property, namely there exists $C \in (0, \infty)$ and an integer $d \in \mathbb{N}$,...
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0answers
48 views

How to obtain the explicit solution of the following integral equation

I'm considering the following integral equation $$ f(x,y,z)=x+\int_0^x\int_0^y\int_0^z f(u,v,w) dudvdw $$ It seems that $$ f(x,y,z)=\sum_{n\geq 0}\frac{x^{n+1} y^n z^n}{(n+1)!n!n!} $$ is the ...
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1answer
47 views

Interchanging limit with series of exponentially growing numbers

Let $(a_i)_{i \in \mathbb{N}}$ be a sequence of positive integers and suppose that $$ \lim\limits_{i \rightarrow \infty} a_i^{1/i} = \mu \in (0, \infty). $$ Consider the function$$ f(\lambda) := \sum\...
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2answers
38 views

Maclaurin Series with $f^{(n)}(0)=0$

I am learning Maclaurin Series for the first time and having trouble understanding it. The thing is, Maclaurin Series has the basic thinking that infinite number of derivatives have coefficients of ...
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1answer
52 views

Question related to radius of convergence

Let $\sum^\infty_{n=0}a_nz^n$ have radius of convergence $r > 0$. Assume that the function $f(z)$ to which it converges has exactly one singular point $z_0$ on $\{z \in \mathbb{C} \vert |z| = r\}$, ...
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2answers
40 views

Doubt about $r$th Term in Binomial theorem

When asked about $10$th term in expansion of $(a+b)^{15}$ we have $$T_{10}=\binom{15}{10}a^5b^{10}$$ But we can also write the binomial as $(b+a)^{15}$ and say $10$th term as $$T_{10}=\binom{15}{10}...
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3answers
64 views

The derivative of $\ln(x)$ [closed]

How can one prove the following by elementary means? $$\ln(x)'=\frac{1}{x}$$ Say we know that $$e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}.$$
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2answers
153 views

Cauchy product of multivariate formal power series

Cauchy product of two univariate formal power series is pretty straight forward. If $$A=\sum_{i=0}^\infty a_i x^i \, ,$$ and $$B=\sum_{j=0}^\infty b_j x^j \, ,$$ then $$A \times B = \sum_{k=0}^\...
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0answers
23 views

Substituting power series in continuity equation in cylindrical form

Following this discussion I'm trying to numerically solve a continuum problem using polynomial power series. Consider the continuity equation in cylindrical form: $$\frac{\partial \rho}{\partial t} + ...
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3answers
142 views

Integral $\int_0^\frac{\pi}{2} \arcsin(\sqrt{\sin x}) dx$

I am trying to calculate $$I=\int_0^\frac{\pi}{2} \arcsin(\sqrt{\sin x}) dx$$ So far I have done the following. First I tried to let $\sin x= t^2$ then: $$I=2\int_0^1 \frac{x\arcsin x}{\sqrt{1-x^4}}...
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0answers
18 views

Expressing series transformed in a certain way

Given $f(t)=a_0+a_1t+a_2t^2+...$, it is straightforward to express $a_0+a_1ut+a_2u^2t^2+...$ through $f$ - it is just $f(ut)$ of course. What I need is to express through $f$ the series$$a_0+a_1t+...
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2answers
17 views

Function Variations of a Power Series Representation

So I am trying to use differentiation to find the power series representations for a series of variations of a function. They are f(x)=1/(8+x)^2 g(x)=1/(8+x)^3 h(x)=x^2/(8+x)^3. I was able to get the ...
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1answer
50 views

Radius of Convergence for $\sum_{n=1}^{\infty}\frac{x^n}{n^{n+1}}$ [closed]

How to find the radius of convergence of the power series $$\sum_{n=1}^{\infty}\frac{x^n}{n^{n+1}}.$$ I've tried finding the $\limsup$ of $|a_n^{1/n}|$ but I'm not having any luck there.
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1answer
23 views

power series and conditional convergence at an end point

Let`s say that $f(x)$ is represented by a power series, and the series is conditionally convergent at the endpoint $x=R+C$, so is $f(x)$ continuous at that point? is it differentiable? since we know ...
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3answers
79 views

How do you power represent $\frac{(1+x)}{(1-x)^2}$?

$f(x)=\frac{(1+x)}{(1-x)^2}$ how do you power represent this formula? I tried it by dividing it into $\frac{1}{(1-x)^2} + \frac{x}{(1-x)^2}$ but could not get a right answer.
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0answers
28 views

The ring $K[[x_1,\ldots,x_n]][x_1^{-1},\ldots,x_n^{-1}]$ is UFD? [duplicate]

It is well known that the ring of formal power series over a field $K[[x_1,x_2,\ldots,x_n]]$ is an UFD. My question is the following: the ring $K[[x_1,\ldots,x_n]][x_1^{-1},\ldots,x_n^{-1}]$ is also ...
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1answer
36 views

Power Series and Recursion

Consider the equation: $$(1-x^2)y'' -xy' +\alpha^2 y=0$$ where $\alpha$ is a real number. For $|x|<1$ and all values of $\alpha$, look for a fundamental set of solutions $y_1$, $y_2$ which are ...
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0answers
20 views

Transform generating function into linear recurrence relation

This problem is inspired from here. Given a generating function $f(x)=\frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are irreducible polynomials, how to transform it into a linear recurrence relation?
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2answers
270 views

Composition of Formal Power Series

Is there a known closed form for the composition of the following power series: $$f(x) = \exp(x) = \sum_k \frac{x^k}{k!}$$ $$g(x) = \frac{1}{1-x} = \sum_n x^n$$ I'd like a closed form power series ...
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2answers
56 views

2 Exercises on Weierstrass M-test for power series

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka 7.29(c),7.30 Please point out errors. Exer 7.29 (c) $M_k = r^k$ because $$\frac{|z|^k}{...
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1answer
211 views

What's the power series of $\log$ at $=1$?

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka (Exer 7.28(a)) Find a power series for $\frac1z$ centred at $z_0=1$. --> I got this. (...
3
votes
0answers
38 views

Matrix functions which are not scalar coefficient power series expansions.

Inspired by this question, I started to wondering (firstly) if there exist any (useful / famous) matrix functions which are not defined as power series expansions with scalar coefficients, but rather ...
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1answer
85 views

Power Series Solutions to Differential Equations $y''-xy=0$

I have the following differential equation $$y''-xy=0\,.$$ By using $y=\sum\limits_{n=0}^{\infty} a_n x^n$, I show that $$(n+2)(n+1)a_{n+2}=a_{n-1}$$ for $n=1,2,...$, and $a_2=0$. (I differentiate the ...
1
vote
2answers
39 views

Manipulation of Power Series [duplicate]

Show that ... $$\sum_{n=1}^\infty \frac{n^2} {2^n} = 6$$ How would one go about in showing that the above power series equals to 6? I would assume that it has something to do with the following ...
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1answer
44 views

Find the region of convergence of the series

Please tell me how to find the region of convergence of following series $$\sum_{n=1}^\infty \frac{(-1)^{n-1} z^{2n-1}}{(2n-1)!}$$ I applied the ratio test for this and got $$\lim_{n \to \infty} \...