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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2
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1answer
997 views

list of convergent series

I wanted to know if there is an online reference I can use to find out known results about convergent series. I could not find this one, for example, on wikipedia $\sum_{k=1}^{+\infty} ...
1
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2answers
117 views

Problem regarding infinite sum of remainders.

Before here @math.SE there was a question regarding a problem on a maths magazine. I decided to look at the link provided, and one problem proposed was (if I'm not recalling this wrongly): Find ...
1
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1answer
257 views

Power Series Definition

What does it mean for a series to be centered around a number? I'm taking complex analysis and am suddenly very confused. I didn't have this explanation, or proof of taylor and power series in ...
2
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1answer
144 views

Divergence of power series in two variables

Let $a_{j,k}:=a_{j+2,k-1} \frac{(j+1)(j+2)}{k}$ for $k>0$ 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 ...
3
votes
1answer
205 views

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 $2<|z|<3$. I try rewriting it in partial fractions as $$ ...
1
vote
1answer
331 views

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 ...
7
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1answer
685 views

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

This is an exercise from Remmert. The power series $\sum\limits_{n=1}^{\infty} \frac {z^{n} }{ n^{2}} \ $ has radius of convergence $1 \ $. Show that the function it represents is injective in $\{ z ...
1
vote
1answer
178 views

Two equations with one solution over infinite variables

Apparently, my previous question didn't get no satisfactory answer, when I asked for two equations having a fixed value for each, not necessarily linear. As XenoGraff states, WolframAlpha does the ...
3
votes
2answers
219 views

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 $|x|<1$. The previous excercise (which was within my limited reach) was to show that ...
5
votes
3answers
144 views

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 ...
8
votes
4answers
671 views

what is the fraction field of $R[[x]]$, the power series over some ring?

I have a question similar to 74335. Let $R$ be an integral domain. Is there a nice description of the fraction field of the power series $R[[x]]$? I know that this field can be a proper subfield of ...
1
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0answers
84 views

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 ...
1
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3answers
160 views

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 ...
4
votes
1answer
2k views

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 $|z| < 1$, find closed forms for the sums $\sum n z^n$ and $\sum n^2 z^n$. My solution: Because $\displaystyle1 + z + z^{2} + ...
1
vote
1answer
531 views

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 $y'=1+y^2$. ...
4
votes
2answers
112 views

For which points is there an analytic function through them

If $(x_i,y_i)$ is an countable infinite set of points, with $x_i\neq x_j$ when $i \neq j$, does there always exist a sequence of real numbers $a_n$, such that $y_i=\sum_{n>1} a_nx_i^n$ for all ...
4
votes
3answers
1k views

Generalised Binomial Theorem Intuition

It was not until recently (why don't they teach it in secondary school?) that I've come across the Generalised Binomial Theorem, which from what I can tell is basically the same as the regular ...
15
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3answers
687 views

Solving a difficult recursion via generating functions

I have been trying to solve the recurrence: \begin{align*} a_{n+1}=\frac{2(n+1)a_n+5((n+1)!)}{3}, \end{align*} where $a_0=5$, via generating functions with little success. My progress until now is ...
10
votes
4answers
418 views

Finding the power series for $y$ where $y + \sin(y) = x$

What do you do to find the power series for an inverse relationship such as for $y$ in $y + \sin(y) = x$? I'm not sure where to begin. (Similarly, the Lambert $W$ function has such a power series ...
0
votes
2answers
681 views

Find the radius of convergence

I am trying to find the radius of convergence for the following function $$ f(x)=\sin(\pi x/4)$$ I already found the Maclaurin series of the function and applied the ratio test but seems I cant get ...
6
votes
2answers
193 views

Power series $x f''(x) + f'(x) + xf(x) = 0$

Find a power series with radius of convergence $R = \infty$ such that $f(x) = \sum_{n=1}^{\infty} a_{n}x^{n}$ satisfies $x f''(x) + f'(x) + xf(x)= 0, \forall \mbox{ } x \in \mathbb R$. How should ...
4
votes
2answers
234 views

Where to find reference about dealing with operators in form of formal power series?

I often encounter the following statements: $${D \over e^D - 1} = {\log(\Delta + 1) \over \Delta}$$ $$\int_x^{x+1} f(t)\,dt= {e^D - 1 \over D} [f]$$ $$\Delta = (e^D - 1)\,$$ $$f(a+x)=e^{a D}[f]$$ ...
0
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2answers
136 views

How to find the Maclurin Series with using the procedure method for $ f(x)=\sqrt{1+2x} $

I stack about the following question Use the procedure method to find the Maclaurin Series for $ f(x)= \sqrt{1+2x} $ Also I would like to know what the procedure method is because I couldn't ...
5
votes
2answers
189 views

How to transform the factored form of $\sin(x)$?

We know $\sin(x)=0$ has solutions $0,\pm\pi,\pm2\pi,\pm3\pi,\dots$. So $\sin(x)$, if interpreted as a polynomial, could be written as: $a_0x^0+a_1x^1+a_2x^2+\cdots$ and we know this polynomial too: ...
1
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4answers
1k views

Sum of reciprocals of square roots

Again, shattered by this question on series, I did have no clue how to begin. Sequences limits are approached through absolute values of the $n$-th term and the assumed limit being smaller than a ...
1
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2answers
60 views

Prove inverse $n$-th powers $< 1/a$ where $a=365$. Find $n$ for the series.

Find $N$ so that for all $n\geq N$ holds that : $$ 2^{-n} + 3^{-n} + 4^{-n} < \frac{1}{365} $$
-1
votes
2answers
6k views

Use a power series to approximate the definite integral to 6 decimal places, help [closed]

Use a power series to approximate the definite integral to 6 decimal places $$\int_0^{0.3} \frac{x^2}{1+x^4} dx$$
0
votes
3answers
103 views

Geometric Series Converges? [duplicate]

Possible Duplicate: Value of $\sum\limits_n x^n$ If I have some real $x$ where $0 < x < 1$ What is the value $y = x + x^2 + x^3 + x^4 + \dots$ ? Intuitively I can see that for $x = ...
4
votes
1answer
253 views

closed form for a series over the Riemann zeta zeros

given the series $ \sum_{\rho} \frac{1}{z-\rho} $ here the sum is taken OVER the roots of the Riemann function on the critical line $ 0 < Re(s) <1 $ the summation is understood as we sum the pair ...
1
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0answers
75 views

series look up site

Is there a site for looking up a series to see some of the associated functions. (In the spirit of Encyclopedia of Integer Sequences OEIS.) In particular I am looking for functions related to $ \sum ...
26
votes
2answers
2k views

Proving an “amazing” claim regarding $\zeta( 3)$ and Apéry's proof

I recently printed a paper that asks to prove the "amazing" claim that for all $a_1,a_2,\dots$ $$\sum_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$$ and thus (probably) ...
1
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1answer
167 views

Formula for $\sum_{k=1}^{n}{k^p}$ where p is a positive integer [duplicate]

Possible Duplicate: why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$ Any hints that can take me from here or am I completely lost. ...
2
votes
1answer
202 views

Best and most efficient way to numerically compute $e$?

There are many well-known methods for efficiently numerically computing $\pi$, such as Chudnovsky's Method or perhaps Gauss-Legendre's algorithm. I was wondering what the best method for computing $e$ ...
1
vote
3answers
111 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 ...
4
votes
1answer
174 views

How can I find power series of $f(x)$?

$$f(x)=\dfrac{1}{1+\dfrac{x}{1+\dfrac{x^2}{1+\dfrac{x^3}{1+\dfrac{x^4}{\ddots}}}}}$$ How can a power series be found given the continued fraction $f(x)$? I'm trying to find $f(x) ...
3
votes
3answers
198 views

Recurrence relations problem (1st order, linear, constant coeff, inhomogeneous)

okay im supposed to find a recurrence relation for $$ a_{n+1}= b \cdot a_n + c \cdot n \ \ \ \ \ \ \ \ \ \ \ \ \mathbf{(1)} $$ where $b$ and $c$ are constants. the method we learned in class was ...
4
votes
3answers
1k views

Can you express $\ln(x)$ as a series with negative or fractional powers of $x$?

I read in my book that $\ln(x)$ cannot be written as a power series. That is a series whose terms contain only non-negative integer powers of $x$. But can $\ln(x)$ be written as an infinite series ...
1
vote
1answer
80 views

What are the properties of functions that cannot be expressed in closed form?

Do they necessarily have asymptotes? Can they be finite over the first interval ($0$ to $x$), infinite over the second ($x$ to $y$), and return to be finite over a third ($y$ to $z$)? When expressed ...
2
votes
1answer
453 views

Given an analytic function on the unit disk, are the coefficients of its series expansion bounded?

Let $D$ denote the unit disk in $\mathbb{C}$. Suppose I have an analytic function $f:D\to D$. Then I can write down its series expansion at, say, $z_0\in D$: $$f(z)=\sum_{n=0}^\infty a_n(z-z_0)^n\,.$$ ...
1
vote
1answer
75 views

Does $|a^{-n} J_0(n)|$ converge?

Does the sequence $|a^{-n} J_0(n)|$ converge? I used the approximation $J_0(n) \approx \sqrt{\frac{2}{\pi n}} \cos(n - \pi/4)$ and assume that $a > 1$. Since the sample sequence of $J_0(n)$ is ...
5
votes
1answer
254 views

The net signed area between $t=0, y=0, t=x$, and $y=f(t)$

f(t) is continuous function.So I know that $\int _0^x {f(t) dt}=$ "The net signed area between $t=0, y=0, t=x$, and $y=f(t)$" And I can find the same result with endless small rectangulars areas ...
5
votes
2answers
196 views

Power series that diverges in specified number of points on the circle of convergence

I want to construct a complex power series with the radius of convergence $R=1$ that diverges in: 1) $k$ points on the circle $\{|z|=1\}$. 2) countable number of points on the circle I have already ...
1
vote
1answer
170 views

How to evaluate $\sum J_0(\alpha n) z^{-n}$ in closed form?

I need to evaluate $\sum_{n = -\infty}^{\infty} J_0(\alpha n) z^{-n}$ in closed form, where $z$ is complex variable and $J_0()$ is the zeroth order Bessel function of the first kind. How do I evaluate ...
0
votes
1answer
126 views

Inquiries concerning the series expansion of the floor function

I'm wondering if an ordinary power series expansion of the floor function is possible, and what it is if it's possible. Additionally, I'm wondering what some of the most common uses of the floor ...
5
votes
1answer
750 views

How does a complex power series behave on the boundary of the disc of convergence?

Until recently, I thought my math teacher had said in his real analysis class that on the boundary of the disc of convergence, a complex power series has at least one point at which it does not ...
3
votes
4answers
558 views

Formula for calculating $\sum_{n=0}^{m}nr^n$

I want to know the general formula for $\sum_{n=0}^{m}nr^n$ for some constant r and how it is derived. For example, when r = 2, the formula is given by: $\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$ ...
2
votes
2answers
473 views

Radius of convergence of power series $\sum_{n=0}^{\infty} \frac{(2x-5)^n}{n^2}$

Consider the below example: $$\sum_{n=0}^{\infty}\frac{(2x-5)^n}{n^2},\qquad c_n=\frac{2^n}{n^2},\qquad R=2^{-1}=\frac{1}{2}.$$ $$\begin{align*} \lim_{n\to\infty}\frac{c_{n+1}}{c_n} &= ...
2
votes
3answers
83 views

Domain of convergence (when not in the correct form)

Radius of convergence is for power series but how does one go about computing the radius of convergence of the infinite sum $$\sum_{k=1}^{\infty} \frac{k}{k+1}\left(\frac{2x+1}{x}\right)^k\ ?$$ Can ...
4
votes
0answers
220 views

Transformation of a Taylor series: “doubling” the derivative order

Suppose a function $f(z)$ has a convergent Taylor expansion: $$f(z)=\sum_{n=0}^{\infty} c_n \frac{z^n}{n!}$$ Are there general tools to compute $$g(z) = \sum_{n=0}^{\infty} c_{2n} ...
1
vote
2answers
84 views

Power series, range

How can we find the first few terms of the Laurent series of $f(w)={1\over \cos w-1}$ where $w\in \mathbb C, |w|<2\pi$? I am wondering if there is another way maybe exploiting one of the more ...