All questions with tag [math: power-series]


convergence of power series

We have given a power series $\sum\limits_{n=0}^\infty a_nx^n$ and $a_0\ne0$. Suppose the power series has radius of convergence $R>0$. Then we may assume that $a_0=1.$ Explanation: It's $$\sum\limits_{n=0}^\infty a_nx^n=a_0\sum\limits_{n=0}^\infty \frac {a_n}{a_0}x^n$$ But why do $\sum\limits_{n=0}^\infty \frac {a_n}{a_0}x^n$ and $\sum\...
2022-07-25 20:46:47

power series expansion of the square root of a Hermitian matrix

Is there a power series development of the square origin of a Hermitian matrix, as a procedure to compute the square origin without taking the inverted or diagonalizing the matrix? I locate for scalar number $x$, $$\sqrt{x}=\sum_{k=0}^\infty \frac{(-1)^k \left((-1+x)^k \left(-\frac12\right)_k\right)}{k!}\qquad\text{for }|-1+x|<1$$, under ...
2022-07-25 20:46:47

Maclaurin expansion of a given function

I am to expand $\ln(2+x)$ as a Maclaurin collection, I've obtained that $\ln(2+x)=\sum\limits_{n=1}^{ \infty}(-\frac{1}{2})^{n}x^{n}$. Can a person examine it?
2022-07-25 20:46:14

power series estimate (convergence)

Let $f(x)=\sum\limits_{n=0}^\infty a_nx^n$ a power series and $f(0)\ne0$. (w.l.o.g. $f(0)=1$) Suppose the power series has radius of convergence $r>0$. A power series is continuous in her convergence interval. So there is a $\delta\in]0,r[$ so that for $|x|<\delta$ it's $|a_1x+a_2x^2+\dots|<1$. My Questions: why is $|a_1x+a...
2022-07-25 20:17:03

Coefficient signs in the sum of successive powers of a polynomial

I'm searching for some structure in the sign variation of the coefficients of: $$P = \sum_{i>0} p^i\enspace,$$ for some polynomial $p \in \mathbb{Z}\langle x\rangle$ with no constant term. I'm interested in whether there is a simple algorithm which, given $n$, will tell me the sign of the coefficient of $x^n$ in $P$. In other words, I'm...
2022-07-25 20:13:59

list of convergent series

I needed to know if there is an on-line reference I can make use of to figure out well-known outcomes concerning convergent collection. I can not locate this set, as an example, on wikipedia $\sum_{k=1}^{+\infty} \left(\frac{1}{2}\right)^k k^2$
2022-07-25 20:05:28

Power Series Definition

What does it suggest for a collection to be focused around a number? I'm taking complex analysis and also am instantly really overwhelmed. I really did not have this description, or evidence of taylor and also power series in calculus, and also I'm assuming below, it outgrew complex analysis and also unreal. Yet, I'm lookin' via guide, tryin' to...
2022-07-25 16:31:30

How can $(z-1)^{-2}(z-2)^{-1}$ be represented as a Laurent series on $2<|z|<3$?

I'm trying to expand $\frac{1}{(z-1)^2(z-2)}$ with $z$ complex on the annulus $2&lt;|z|&lt;3$. I try rewriting it in partial fractions as $$ \frac{1}{(z-1)^2(z-2)}=-\frac{1}{z-1}-\frac{1}{(z-1)^2}+\frac{1}{z-2}. $$ I know I can make the last summand above converge on $|z|&gt;2$, by writing it as $\frac{1}{z}\cdot\frac{1}{1-2/z}$. How...
2022-07-25 16:09:15

Problem regarding infinite sum of remainders.

Before below @math. SE there was an inquiry pertaining to a trouble on a mathematics publication. I determined to consider the link given, and also one trouble recommended was (if I'm not remembering this mistakenly) : Find $$\sum\limits_{n = 0}^\infty {{{\left( { - 1} \right)}^n}\left[ {f\left( x \right) - {T_n}\left( x \right)} \right]} $$ ...
2022-07-25 15:58:02

Divergence of power series in two variables

Let $a_{j,k}:=a_{j+2,k-1} \frac{(j+1)(j+2)}{k}$ for $k&gt;0$ and $a_{j,0}:=(-1)^j$, thus $a_{j,k}=(-1)^j\frac{(2k+j)!}{j!k!}$, $j,k\geq 0$. Now, I have to show that the series defined by $S=\sum_{j,k}a_{j,k}x^j y^k$ diverges for any value except for $(x,y)=(0,0)$. Probably, there is some analogous formula for the radius of convergence of a p...
2022-07-25 15:49:41

The power series $\sum\limits_{n=1}^{\infty} \frac {z^{n} }{ n^{2}} \ $

This is a workout from Remmert. The power series $\sum\limits_{n=1}^{\infty} \frac {z^{n} }{ n^{2}} \ $ has distance of merging $1 \ $. Show that the function it stands for is injective in $\{ z \in \mathbb{C} | \ \ \lVert z \rVert &lt; \frac{2}{3} \} \ $. The message offers the tip: $z^n -w^n = (z-w)\ ( z^{n-1}+z^{n-2}w + \ldots + w^{n-1...
2022-07-25 10:53:48

Series Solution Near Ordinary Points for Second Order Differential Equations

Given $(1+x^2)y''+2xy'-2y = 0$ The above equations obviously has analytic points everywhere except for $x=1$ and $-1$. Find two linearly independent solutions $y_1$ and $y_2$ to the differential equation valid near $x_0=0$. To make life a little easier, choose the linearly independent equations: $y_1$ $:$ $a_0$ = $y(x_0)$ = 1 and $a_1$ = $y'(x_...
2022-07-25 10:41:09

Manipulation of some power series (probably integration or derivation).

Show that $\ln\Big(|\frac{1+x}{1-x}|\Big)=2\sum_{n=0}^{\infty}\frac{x^{2n+1}}{2n+1},$ for $|x|&lt;1$. The previous excercise (which was within my minimal reach) was to show that $\frac{1}{1-x^2}=\sum_{n=0}^{\infty}x^{2n},$ for $|x|&lt;1$. I believe there is a (not extremely refined) link below yet, obviously, I can not see it. I do not r...
2022-07-25 10:38:42

Two equations with one solution over infinite variables

Apparently, my previous inquiry really did not get no sufficient solution, when I requested for 2 formulas having actually a dealt with value for each and every, not always straight . As XenoGraff states, WolframAlpha does the job, yet counts permutations of values amongst variables, and also is hence not practical to examine any kind of 2 formu...
2022-07-25 10:36:51

Is this action of $\mathbb F[[x]]$ on $\bigoplus_{i=0}^{\infty}\mathbb F$ natural?

The title of my question has a field $\mathbb F$ in it, but to make sure I'm not losing anything, I would like to introduce my question in full generality. But still, I will be happy with an answer in fields, which is why I chose this title. Please feel free to consider $R$ a field and all modules to be vector spaces. Notation I'm assuming $\mat...
2022-07-24 09:35:22

Squaring an arbitrary summation?

I'm trying to find a recurrence relation for the coefficients for the Maclaurin series for $\tan(x)$ by substituting $y=\sum_{k=0}^{\infty}C_{2k+1}x^{2k+1}$ into the differential equation $y'=1+y^2$. This is because $\tan(x)$ is the solution to the initial value problem for the aforementioned DE with the initial condition $y(0)=0$; this is also ...
2022-07-24 06:25:54

subrings A of the ring of power series k[[t]] with the condition (A : k[[t]]) $\neq$0 and k $\subset$ A

I would like to understand the structure of the subrings A of the ring of formal power series k[[t]] (where k is a field) which satisfy the condition (A : k[[t]]) $\neq$ 0 and k $\subset$ A. Are they of the form v(a)$\geq$ n + k ? v is the order of the formal power series and (A : k[[t]]) = a k[[t]] $\subset$ A. Would someone help me with that?
2022-07-24 06:09:29

What is the fraction field of $R[[x]]$, the power series over some integral domain?

I have an inquiry comparable to 74335. Allow $R$ be an integral domain. Exists a wonderful summary of the portion area of the power series $R[[x]]$? I recognize that this area can be a correct subfield of $\operatorname{Frac}(R)((x))$, the Laurent collection over the portion area of $R$, as seen here. Considered that, I'm at a loss of variou...
2022-07-24 06:07:16

Finding closed forms for $\sum n z^{n}$ and $\sum n^{2} z^{n}$

Using the identity $\frac{1}{1-z} = 1 + z + z^2 + \ldots$ for $|z| &lt; 1$, find closed forms for the sums $\sum n z^n$ and $\sum n^2 z^n$. My solution: Because $\displaystyle1 + z + z^{2} + \ldots = \frac{1}{1-z}$, $\displaystyle1 + 2z + 3z^2 + \ldots = \sum_{n=1}^\infty n z^{n-1} = \frac{1}{(1-z)^2}$ and $\displaystyle\sum_{n=1}^\infty n...
2022-07-24 06:00:10

For what values of x does the series $1+\frac{x}{3}+\frac{x^2}{5}+\frac{x^3}{7}+\cdot\cdot\cdot$ converge?

For what values of x does the series $1+\frac{x}{3}+\frac{x^2}{5}+\frac{x^3}{7}+\cdot\cdot\cdot$ converge? The solution states: The general term is of the form $u_n(x)=\frac{x^{n-1}}{(2n-1)}$, and hence $$\frac{|u_{n+1}|}{|u_n|}=\frac{|x^n|}{(2n+1)}\cdot\frac{(2n-1)}{|x^{n-1}|}$$ ------edit start------- $$=\frac{(2n-1)}{(2n+1)}\cdot\frac{|x^n|}{...
2022-07-24 05:33:55