Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n x^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex (or more generally an element of a Banach algebra).

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3answers
112 views

Formula to $\ln$ that holds on interval $x \geq 1$

In the Wikipedia we can find two formulas using power series to $\ln(x)$, but I would like a formula that holds on the interval $x \geq 1$ (or is possible to calculate $\ln(x)$ to $x \geq 1$ with the ...
0
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0answers
23 views

Convolution vs term wise product

Fix an element $x\in\mathbb R^\times$. Then I can define the ring $\mathbb R[[x]]$ which is the set: $$\left\{\sum_{i=0}^\infty a_ix^i\,: a_i\in\mathbb R\right\}$$ and where the product is the usual ...
0
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1answer
57 views

How to solve $xy''+2y'+\lambda^2 \, xy=0$ with the power series method?

Find all functions $y(x)$ which can be expressed as a convergent power series $y(x)=\sum_{n=0}^{\infty} a_n x^n$ and which satisfy the following differential equation: $$xy''+2y'+\lambda^2 \, xy=0$$
0
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1answer
54 views

Prove that $ \frac12 [log(1-x)]^2 = \frac12x^2 + (1+ \frac12) \frac13 x^3+\cdots$ for $-1<x<1$

$\frac12 [log(1-x)]^2 = \frac12 x^2 + (1+ \frac12) \frac13 x^3+ (1+ \frac12 + \frac13) \frac14 x^4+ \cdots$ My attempt: I'm thinking of finding a series which is convergent in $-1<x<1$ and ...
0
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1answer
134 views

Find the radius of convergence of the power series

$\displaystyle\sum_{n=0}^{\infty}a_nz^n$, where $a_{2k+1} = 2^k$ and $a_{2k} = (1 + (1/k))^2$ for $k = 0, 1, 2, \dotsc$ I started off by doing the ratio test, but I know that the ration test is for ...
0
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1answer
169 views

What's the limit of coefficient ratio for a reciprocating power series?

I have a question about the coefficient in the inverse of the power series. Assume $$ f=1-\sum_{i=1}^{\infty}(ck_i)x^i, $$ where $c$ and $k_i$ are positive and $0<ck_i<1$ for any $i>0$. ...
0
votes
1answer
106 views

How can we take a power series and multiply each term, i.e. $c_n x^n$ by $y^n$?

In other words, given a power series $f(x)$, is there an alternative to taking $\lim_{x\to{x y}}f(x)$? I ask this because I thought that there may be a way to replace the limit by integration, or ...
0
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2answers
90 views

Trying to revert a series with problematic log term

I'm stuck on a problem which I'm not sure has a solution. I have the first few terms of a series I want to invert, $y(x)=\ln(x)+a_0+\frac{a_1}{x}+\frac{a_2}{x^2}+\cdots$ I know the inverse exists ...
-2
votes
1answer
52 views

Find the interval of convergence for these 3 power series

I believe I need to use the root test and ratio test. I've solved the first two, but I'm not too sure I fully understand how to do these, so was hoping someone else could work them out so I could ...
8
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4answers
345 views

Asymptotic behavior of $\sum\limits_{k=1}^n \frac{2^k}{k}$

I'm looking for an asymptotic equivalent of $$\sum_{0 < k \le n} \frac{2^k}{k}$$ as $n \to \infty$. A plausible candidate seems to be $\frac{2^{n+1}}{n+1}$ (WolframAlpha plot, and the intuitive ...
4
votes
3answers
215 views

Power Series $0^{0}$

My textbook explains that the power series: $\sum_{n=0}^{\infty} x^{n}/n!$ converges for $x=0$ because the terms of the series get the value 0. My problem with this argument is the first term, ...
3
votes
3answers
96 views

Puiseux Series?

WolframAlpha says that $$\sqrt{x^2-1}$$ expanded in Puiseux series near 1 is $\sqrt 2 \sqrt{x-1}$ I don't know what is the Puiseux series, I have search on the net but I don't have understood so ...
3
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0answers
64 views

Find a series solution to $(x^2-2)y''+6xy'+4y=0$.

Find a series solution to $(x^2-2)y''+6xy'+4y=0$. A. Find the recurrence relation to $a_n$: My answer is $a_{n+2}=a_n\cdot \frac{n+4}{2(n+2)}$ which is correct. B. Using A, write two independent ...
3
votes
4answers
238 views

Show that $\displaystyle\lim_{n\to\infty} nx^n=0$ for $x\in[0,1)$.

Show that $\displaystyle\lim_{n\to\infty} nx^n=0$ for $x\in[0,1)$.
3
votes
2answers
755 views

Why is Taylor series expansion for $1/(1-x)$ valid only for $x \in (-1, 1)$?

After finding an expansion of $$ \frac{1}{1-x} = 1 + x + x^2 + x^3 + \ldots$$ a quick test of various values for $x$ reveals that this expansion is not valid for $\forall x \in \mathbb{R}-\{1\}$. ...
3
votes
7answers
352 views

how to find this generating function

this is the power series: $$\sum_{i=0}^\infty n(n-1)^2 (n-2) z^n.$$ how can I find a generating function from it? I could use the third derivative but the $n-1$ is squared so I don't know what to ...
2
votes
2answers
119 views

What is the expansion in power series of ${x \over \sin x}$

How can I expand in power series the following function: $$ {x \over \sin x} $$ ? I know that: $$ \sin x = x - {x^3 \over 3!} + {x^5 \over 5!} - \ldots, $$ but a direct substitution does not give me a ...
2
votes
2answers
60 views

Sum of powers of 2 from 1 to log(N).

I came across the following sum: $\sum_{m=1}^{\log_2(N)} 2^{m}$. My intuition tells me that this should be bounded by 2N, but how would I prove this?
2
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2answers
464 views

Solution to ODE using Power Series

I'm currently trying to wrap my head around how to solve an ODE with series. The problem I am working on is this: Find the indicated coefficients of the power series solution about x=0 of the ...
2
votes
1answer
149 views

If $f$ is analytic, prove that $\overline{f(\overline{z})}$ is also analytic

Let $f$ be an analytic function in an open set $U \subseteq \mathbb{C}$. Let $V=\{z\in\mathbb C:\overline z\in U\}$. Define $g$ on $V$ by $g(z)=\overline{f(\overline{z})}$. Show that $g$ is analytic ...
2
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2answers
122 views

Calculate Laurent Series for $\frac{\ln z}{(z-1)^3}$ about $z=1$

Calculate the Laurent series of the function $g(z)= \frac{\ln z}{(z-1)^3}$ about the point $z=1$. Well since the singularity and the centre of the circle we are expanding about collide, I can just ...
2
votes
1answer
116 views

Evaluate $ \int_0^1 \sum_{k=0}^\infty (-x^4)^k dx = \int_0^1 \frac{dx}{1+x^4} $

I have read this thread and I found in some comments the above named equality. I couldn't follow the transformation, which are done to get from the left to the right side at that point in particular. ...
2
votes
1answer
64 views

Does n power of e grow much more faster than its Maclaurin polynomial? [duplicate]

I wonder how to calculate the following limit: $$ \lim_{n\rightarrow\infty}\frac{1+n+\frac{{}n^{2}}{2!}+\cdots +\frac{n^{n}}{n!}}{e^{n}} $$ In the first sight, I think it should be zero, because ...
2
votes
1answer
114 views

What's $\sum{\frac{x^n}{n^3}}$?

What's $\displaystyle f(x)=\sum_{n=1}^\infty{\frac{x^n}{n^3}}$? Note its derivative: $$\displaystyle f'(x)=\sum_{n=1}^\infty{\frac{x^{n-1}}{n^2}}$$ and the next derivative: $$\displaystyle ...
2
votes
2answers
76 views

How can i evaluate this power series?

$\sum_{n=0}^{\infty }\frac{1}{2n+1} \left (\frac{1}{3} \right )^{n}\left ( -1 \right )^{n} $ it's solved by power series of arctan. is it possible the answer written by real number?
2
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3answers
54 views

A power series from $\frac{x}{9+x^2}$

I need to make power series from $\frac{x}{9+x^2}$, and I don't have any idea how. The only thing I know is how to make power series from $\frac{1}{1-q}$. Thank you!
2
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1answer
234 views

Definite Sum of Confluent Hypergeometric involving power function

I find it difficult to evaluate the following definite sum: $$ \sum _{k=1}^K \frac{_1F_1[k,1,x]} {2^k} $$ Thank you for your time
2
votes
2answers
76 views

An approximate solution to an ODE

I am interested in the ODE: $x^\prime = x^2 + t^2$ $x(0)=0$ The power-series method is not (easily?) applicable here. Do you have any suggestions how to solve it?
2
votes
3answers
257 views

Showing that $R(x)$ is a proper subset of $R((x))$ if R is a field

I would like to show that if $R$ is a field, then $R(x)$ is a proper subset of $R((x))$, where $R(x)$ is the ring of rational functions, and $R((x))$ is the ring of formal Laurent series. If $f \in ...
2
votes
3answers
111 views

If $a_0\in R$ is a unit, then $\sum_{k=0}^{\infty}a_k x^k$ is a unit in $R[[x]]$

Let $R$ a ring, and let $$\displaystyle R[[x]]=\left\{\sum_{k=0}^{\infty}a_k x^k\;\middle\vert\; a_k\in R\right\}$$ with addition and multiplication as defined for polynomials. We have that $R[[x]]$ ...
1
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1answer
94 views

Statistical problem: how many books of different widths fit it into a self of a limited certain width?

Let's say I have N sets of books, being the size of the books in a set the same. The cardinality of the every set is different: I might have 3 books of width 5 units (first set), 6 books of width 10 ...
1
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4answers
72 views

Factorial in power series; intuitive/combinatorial interpretation?

It is well known that the terms of the power series of exponential and trigonometric functions often involve the factorial function, essentially as a consequence of iterating the power rule. My ...
1
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1answer
295 views

Radius of convergence of a power serise involving the Fibonacci sequence.

Consider the power series $$\sum_{n=0}^{\infty}a_nz^n.$$ where, $a_0=0$ , $a_1=1$ , $a_n=a_{n-1}+a_{n-2}$. Find the radius of convergence of the power series. MY Attempt : Clearly $\{a_n\}$ is a ...
1
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0answers
122 views

Infinitely nested radicals

In a recent paper it was stated (and maybe proved) that we can solve any polynomial equation with nested radicals. Here "nested radicals" means expression such as: $$ ...
1
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1answer
57 views

Exponential of a complex number converges absolutely

$$\operatorname{exp}(z)=\sum_{n=0}^\infty \frac{z^n}{n!}$$ This converges absolutely for every $z\in \Bbb C$. What does this mean to a layman?
1
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1answer
50 views

Finding the coefficients of $h(z)$ laurent series

Consider: $$h(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$$ Find the coefficients $a_n$ of the Laurent Series of $h(z)$ centered at $z=-2$ I got this from the approach here: Infinite sum complex analysis ...
1
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1answer
176 views

Derive an explicit formula for a power series

Could anyone help me find an explicit formula for: $$ \sum_{n=1}^\infty n^2x^n $$ We're supposed to use: $$\sum_{n=1}^\infty nx^n = \frac{x}{(1-x)^2} \qquad |x| <1 $$
1
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1answer
287 views

Calculating the power series expansion about $\pi/2$ of $g(z)=\tan(z/2)$

Calculate the power series expansion about $\pi/2$ of $g(z)=\tan(z/2)$. Now calculate the expansion about $0$. I'm having trouble doing this. I'm not even sure which is the best way to approach ...
1
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4answers
133 views

power series for square root matrix

Suppose I have a matrix of the form $$U\ =\ (I+z\thinspace X)^{\frac{1}{2}}$$ where $I$ is the $n\times n$ identity matrix, $z\in\mathbb{C}$ and $X$ is a $n\times n$ arbitrary complex matrix with ...
1
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2answers
90 views

Matrix Inversion Test ( Sum of Matrix series)

Friends,I have a set of matrices of dimension $3\times3$ called $A_i$. , Following are the given conditions a) each $A_i$ is non invertible except $A_0$ because their determinant is zero. b) ...
1
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2answers
2k views

Solution of the Legendre's ODE using Frobenius Method

This is the Legendre's differential equation given in my book: $(1-x)^{2}\ddot{y}-2x\dot{y}+k(k+1)y=0$ I solved this equation by taking: $y=x^{c}\{a_{0}+a_{1}x+a_{2}x^{2}+.....+a_{r}x^{r}+.....\}$ ...
1
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1answer
234 views

Introduction to Analysis: Multiplication Theorem for Series

I've been stuck on this problem over the weekend so I decided to ask for some direction. The problem reads: "The multiplication theorem for series requires that the two series be absolutely ...
1
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0answers
62 views

Solving ODE with negative expansion power series [duplicate]

I am solving a series of ODE, such that each DE is equal to some degree of term that I'm expanding to. For instance, one DE is this: $\xi^r\partial_r g_{rr}+2g_{tt}\partial_t\xi^t=\mathcal{O}(r)$ ...
1
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1answer
130 views

Modelling ellipsoid-like surface with equations

In my numerical simulation, I would often have to generate the failure surface of some materials. Often, the failure surface will take a ellipsoid-like ball such as this: Of course, the surface ...
1
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1answer
287 views

using power series expansion to find a holomorphic function which solves a differential equation

Using power series expansions, find a function $f$ which is holomorphic on the unit disk $D:=$ {$z\in\mathbb C:|z|<1$} and solves the differential equation $(1-z^2)f''(z)-4zf'(z)-2f(z)=0$ for ...
1
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1answer
2k views

Radius of convergence for the exponential function

I'm studying physics and am currently following a course on complex analysis and in the section on analytic functions, the radius of convergence $R$ for power series was introduced. The Taylor ...
0
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0answers
50 views

Factorial moment and derivative of probability generating function

The $k$-th factorial moment of a non-negative integer random variable $X$ is given by the expectation $E[(X)_k] = E[X(X-1)\cdots(X-k+1)]$. The probability generating function of the same $X$ is given ...
0
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1answer
85 views

Taking Limits of Sets

I know this sounds like a ridiculous idea- but it's the only one I can think of for this radius of convergence problem for a power series involving sine. I want to let $P:= \{k:|sin(k)| \geq \delta ...
0
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0answers
37 views

Question about the coefficient of operator

Note that the "coefficient of" operator is an operator that takes the coefficient of the power series. We start with the following: $$ \frac{1}{f(x)+z} - \frac{1}{f(x)} = \sum_{k=0}^\infty ...
0
votes
1answer
66 views

Use series methods to find solution corresponding to..

Use series methods to find solution corresponding to $a_0 = 1$ for the equation $(x+1)y' - y = 0$ Here is my work. Can someone verify that I have the correct solution: So for my final solution I ...