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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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How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

How can I evaluate $$\sum_{n=1}^\infty \frac{2n}{3^{n+1}}$$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
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I'd like to determine the function corresponding to the following power series: $$x + \sum_{n=1}^\infty (-1)^n\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots2n} \frac{x^{2n+1}}{2n+1},$$ where $|... 2answers 6k views Sum of a power series$n x^n$[duplicate] I would like to know: How come that $$\sum_{n=1}^\infty n x^n=\frac{x}{(x-1)^2}$$ Why isn't it infinity? 4answers 1k 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:$\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$... 3answers 692 views Find the form of a second linear independent solution when the two roots of indicial equation are different by a integer Consider the differential equation $$x^2y''+3(x-x^2)y'-3y=0$$$(a)$Find the recurrence equation and first three nonzero terms of the series solution in powers of $$corresponding to the larger root ... 4answers 2k views Formal power series coefficient multiplication Given that I have two formal power series:$$ A(x) = \sum_{k \ge 0} a_k x^k  B(x) = \sum_{k \ge 0} b_k x^k $$The Cauchy Product gives a series$$ C(x) = \sum_{k \ge 0} c_k x^k  c_k = \... 3answers 781 views How to calculate$f(x)$in$f(f(x)) = e^x$? How would I calculate the power series of$f(x)$if$f(f(x)) = e^x$? Is there a faster-converging method than power series for fractional iteration/functional square roots? 3answers 764 views What is the Riemann surface of$y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$? The following appears as the second-to-last problem of Stewart's Complex Analysis: Describe the Riemann surface of the function$y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$. This problem ... 1answer 1k views Deriving Maclaurin series for$\frac{\arcsin x}{\sqrt{1-x^2}}$. Intrigued by this brilliant answer from Ron Gordon, I was attempting to find the Maclaurin series for $$f(x)=\frac{\arcsin x}{\sqrt{1-x^2}}=g(x)G(x)$$ with$g(x)=\frac{1}{\sqrt{1-x^2}}$and$G(x)$... 2answers 178 views How to find the radius of convergence? The function is$\dfrac {z-z^3}{\sin {\pi z}} $. How to find the radius of convergence in$ z=0 $? 5answers 368 views Identify infinite sum:$\sum\limits_{n=0}^{+\infty}\frac{x^{4n}}{(4n)!}$Find$f(x)$, the unknown function satisfying $$f(x) = \sum\limits_{n=0}^{+\infty}\frac{x^{4n}}{(4n)!}$$ I'm looking for a direct solution which is different from mine, if possible. 2answers 4k views Radius of convergence of a sum of power series I have two series$\displaystyle\sum_{n=1}^{\infty} a_n x^{n}\displaystyle\sum_{n=1}^{\infty} b_n x^{n}$with radius of convergence$2$and$3$respectively. How can I find the radius of ... 2answers 2k views Proof that$\sum\limits_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$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) ... 1answer 502 views Why$e^{\pi}-\pi \approx 20$, and$e^{2\pi}-24 \approx 2^9$? This was inspired by this post. Let$q = e^{2\pi\,i\tau}$. Then, $$x := \left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24} = \frac{1}{q} - 24 + 276q - 2048q^2 + 11202q^3 - 49152q^4+ \cdots\tag1$$ ... 4answers 170 views Bernoulli Number analog using Cosine I know that Bernoulli Numbers can be found with the generating function $$\frac{x}{e^x-1}=\sum_{n=0}^{\infty}\frac{B_n}{n!}x^n$$ I was wondering if any work has been done using a similar equation $$\... 1answer 635 views Riemann Zeta Function Manipulation The Riemann zeta function is defined on the Re z> 1 by$$\zeta(z)=\sum_{n=1}^\infty \frac{1}{n^z}$$(i) show that for Re z> 1, we have$$(1-2^{1-z})\zeta(z)=\sum_{n=1}^\infty \frac{(-1)^{n+... 1answer 244 views Uniform convergence in the endpoints of an interval Study the pointwise and uniform convergence of the series $$\sum_{n=1}^\infty\dfrac{4^n}{n^2}\dfrac1{(1+x^2)^n}$$ I'm doing this exercise and I'm not sure about the following: What I've done is ... 1answer 188 views Lagrange Bürmann Inversion Series Example I am trying to understand how one applies Lagrange Bürmann Inversion to solve an implicit equation in real variables(given that the equation satisfies the needed conditions). I have tried looking for ... 3answers 260 views Summing up the series$a_{3k}$where$\log(1-x+x^2) = \sum a_k x^k$If$\ln(1-x+x^2) = a_1x+a_2x^2 + \cdots \text{ then } a_3+a_6+a_9+a_{12} + \cdots = $? My approach is to write$1-x+x^2 = \frac{1+x^3}{1+x}$then expanding the respective logarithms,I got a series (... 1answer 1k views Is it possible for a function to be smooth everywhere, analytic nowhere, yet Taylor series at any point converges in a nonzero radius? It is well-known that the function $$f(x) = \begin{cases} e^{-1/x^2}, \mbox{if } x \ne 0 \\ 0, \mbox{if } x = 0\end{cases}$$ is smooth everywhere, yet not analytic at$x = 0$. In particular, its ... 4answers 1k views How to express$(1+x+x^2+\cdots+x^m)^n$as a power series? Is it possible to express$(1+x+x^2+\cdots+x^m)^n$as a power series? 2answers 657 views How to do a very long division: continued fraction for tan I want to compute $$\tan(r) = \cfrac{r}{1 - \cfrac{r^2}{3 - \cfrac{r^2}{5 - \cfrac{r^2}{7 - {}\ddots}}}}$$ by dividing the power series for sin and cos as it is said can be done in http://arxiv.org/... 1answer 148 views Proving that$\lim_{x\to1^-}\left(\sqrt[a]{1-x}\cdot\sum_{n=0}^\infty~x^{n^a}\right)=\Gamma\left(1+\frac1a\right)$How could we prove that $$\lim_{x\to1^-}~\bigg(\sqrt[a]{1-x}\cdot\sum_{n=0}^\infty~x^{n^a}\bigg)~=~\Gamma\bigg(1+\frac1a\bigg)$$ for$a>0$? The inspiration came to me while trying find a ... 3answers 707 views Cauchy product on exponential-looking power series Original posting by dioxen here: Double summation including power and factorial I am finding some trouble in computing the following sum: $$\sum_{k=0}^\infty \frac{x^k}{k!}\;\sum_{m=0}^k\frac ... 2answers 510 views Series of nested integrals I'm trying to calculate the following series of nested integrals with \varepsilon(t) being a real function.$$\sigma = 1 + \int\nolimits_{t_0}^t\mathrm dt_1 \, \varepsilon(t_1) + \int_{t_0}^t\... 1answer 801 views Radius of convergence of power series Given a meromorphic function on$\mathbb{C}$, is the radius of convergence in a regular point exactly the distance to the closest pole? As Robert Israel points out in his answer, that this is of ... 2answers 439 views The value of a limit of a power series:$\lim\limits_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \left(\frac{x}{k} \right)^k$What is the answer to the following limit of a power series? $$\lim_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \left(\frac{x}{k} \right)^k$$ 3answers 236 views Series of inverses of binomial coefficients Can you think of a simple way of proving that $$\sum_{n=k+1}^\infty \frac{1}{n \choose k}$$ is rational for any$k \geq 2$? Here's the background. Consider a series: $$\sum_{n=1}^\infty \frac{1}{... 2answers 631 views \lim\limits_{x\to\infty}f(x)^{1/x} where f(x)=\sum\limits_{k=0}^{\infty}\cfrac{x^{a_k}}{a_k!}. Does the following limit exist? What is the value of it if it exists?$$\lim\limits_{x\to\infty}f(x)^{1/x}$$where f(x)=\sum\limits_{k=0}^{\infty}\cfrac{x^{a_k}}{a_k!} and \{a_k\}\subset\mathbb{N} ... 6answers 505 views Why is \sum_{n=0}^{\infty}\frac{x^n}{n!} = e^x? I am trying to see where this relationship comes from: \displaystyle \sum_{n=0}^{\infty}\frac{x^n}{n!} = e^x Does anyone have any special knowledge that me and my summer math teacher doesn't know ... 1answer 245 views Why don't we have an isomorphism between R[x] and R[[x]]? R is a ring. Why don't we have an isomorphism$$ R[x] \cong R[[x]]\ ? $$3answers 1k views Taylor series for different points… how do they look? I can't understand what it means to do the Taylor series at the point a. The best way would be showing me how it looks for different a on a graph. Do I find those graphs on the Internet? 3answers 242 views Could we show 1-(x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dots)^2=(1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}- \dots)^2 if we didn't know about Taylor Expansion? Suppose that humanity haven't discovered Taylor Series Expansion of trigonometric functions or of any function that would help us on this. Which means we are not allowed to replace the given infinite ... 1answer 203 views How to calculate the integral of x^x between 0 and 1 using series? [duplicate] How to calculate \int_0^1 x^x\,dx using series? I read from a book that$$\int_0^1 x^x\,dx = 1-\frac{1}{2^2}+\frac{1}{3^3}+\dots+(-1)^n\frac{1}{(n+1)^{n+1}}+\cdots$$but I can't prove it. Thanks in ... 2answers 1k views What is the radius of convergence of \sum z^{n!}? How to find the radius of convergence of \sum z^{n!}? I'm used to applying the ratio test to power series of the form \sum a_{n}z^{n}, but for a different power of z, I am a bit stumped. What ... 1answer 200 views Is there a generalization of the fundamental theorem of algebra for power series? Given the similarity between polynomials and power series, I was wondering if there is any generalization of the fundamental theorem of algebra for power series. I understand that it doesn't make much ... 1answer 173 views Prove the following equation of complex power series. Show that for |z| \lt 1 with z \in \Bbb C, we have$$ \sum_0^\infty \frac{{z^2}^k}{1-{z^2}^{k+1}} = \frac{z}{1-z}  \sum_0^\infty \frac{2^k{z^2}^k}{1+{z^2}^{k}} = \frac{z}{1-z} $$My guess ... 5answers 1k views How to create alternating series with happening every two terms I'm looking for a technique for creating alternating negatives and positives in a series. Specifically: when n=1, the answer is +, n=2 is +, n=3 is -, n=4 is -... etc. I have every other part of the ... 1answer 155 views How to estimate the growth of a “savage” function near 1? Say I have a function which exists within the unit disk, say$$f(x)=a_0+a_1x+a_2x^2+...$$If we know sufficient information about the coefficients, say we know the growth rate of \sum\limits_{k=0}^{n}... 1answer 119 views Proving the inequality |e^z-1|\leq e^{|z|}-1 I am trying to prove this inequality$$|e^z-1|\leq e^{|z|}-1\leq |z|e^{|z|}$$I've tried calculating the difference in their power series$$\sum_{k=0}^\infty\frac{|z|^k}{k!}-1-\left|\,\sum_{k=0}^\... 3answers 232 views Evaluate this power series Evaluate the sum $$x+\frac{2}{3}x^3+\frac{2}{3}\cdot\frac{4}{5}x^5+\frac{2}{3}\cdot\frac{4}{5}\cdot\frac{6}{7}x^7+\dots$$ Totally no idea. I think this series may related to the$\sin x$series ... 3answers 92 views Compute the sum$\sum_{k=1}^{\infty}k^mz^k$where$|z|<1$Do you know how to find the limit of$\sum_{k=1}^{\infty}k^mz^k$where$|z|<1$and m is a natural number? I've tried to google it in wiki but I do not understand the closed form (http://en.... 2answers 453 views How to construct this Laurent series? How do I construct the following Laurent series (clipped off Wolfram Alpha)? I know that the numerator can be written as$-1+\frac{\pi}2 z-...$Alternatively (without the Laurent series), how can I ... 3answers 113 views Series$\frac{x^{3n}}{(3n)!} $find sum using differentiation Find sum of the series $$\sum_{n=1}^{\infty}\frac{x^{3n}}{\left(3n\right)!}$$ using differentiation. So far I found that $$S(x)+1=S'''(x)$$ but it does not help. Then I tried different interesting ... 1answer 55 views Does the Abel sum 1 - 1 + 1 - 1 + … = 1/2 imply$\eta(0)=1/2$? If$\sum_{n=1}^\infty a_n$is Abel summable to$A$, then necessarily$\sum_{n=1}^\infty a_n n^{-s}$has a finite abscissa of convergence and can be analytically continued to a function$F(s)\$ on a ...
There are two identities that have a seemingly dual correspondence: $$e^x = \sum_{n\ge0} {x^n\over n!}$$ and $$n! = \int_0^{\infty} {x^n\over e^x}\ dx.$$ Is there anything to this comparison? (I ...