So i need in finding the correct Taylor series(expansion) for the following function and use it in calculating the sum of series:  $$f(x)=\ln(\frac{1-4x^2}{1+4x^2})$$  $$\sum_{n=1}^{\infty}\frac{1}{(2n-1)4^{2n-1}}$$  so i did this:  $$f(x)=\ln(\frac{1-4x^2}{1+4x^2})=\ln(\frac{1+4x^2-4x^2-4x^2}{1+4x^2})$$ $$\ln(1+\frac{-8x^2}{1+4x^2})$$  and then i used this :  $$\ln(1+x)=\sum_{n=1}^{\infty}\frac{(-1)^n x^{n+1}}{n+1}$$ for $$x\in(-1,1]$$  and then i got that:  $$\sum_{n=1}^{\infty}\frac{(-1)^n (\frac{-8x^2}{1+4x^2})^{n+1}}{n+1}$$  and i modified the bounds for x.  Firstly, i don't really know if foudn the correct taylor series, and secondly it does not really help with caculating sum. Help ragarding both taylor series and caculating sum would be appreciated. Thank you in advance.
What is the Taylor series of $$\ln \frac{1-x^2}{1+x^2}$$   ?   I started by evaluating the first derivatives but the more I go, the more complicated they are and I can't identify a pattern.
Suppose one is trying to find the radius of convergence of  $$ \exp(x)=\sum\limits_{k=0}^{\infty} \frac{x^k}{k!} $$  This "proof" was given in the lecture: $$ \begin{align} r =&amp; \frac{1}{\limsup\limits_{n \rightarrow \infty}\sqrt [n]{|a_n|}} \\ =&amp; \frac{1}{\limsup\limits_{n \rightarrow \infty}\sqrt [n]{\frac{1}{n!}}} \\ =&amp;\lim\limits_{n \rightarrow \infty} \sqrt [n]{n!}\\ r=&amp; \infty \end{align}$$  Other proofs involving the ratio test also seem to inspect the convergence of $\frac{1}{k!}$ instead of $\frac{x^k}{k!}$. Why is one allowed to substitute $\frac{1}{k!}$? What Does it Really Mean to Have Different Kinds of Infinities? Can someone explain to me how there can be different kinds of infinities?  I was reading "" by  and came across the concept of countable and uncountable infinities, but they're only words to me.  Any help would be appreciated. How to check if installing a package will require a reboot? I want to search for all .conf files in /etc/. I tried grep -r *.conf /etc, but the result isn't right. What am I doing wrong? ACPI error when trying to make dual boot with Ubuntu 18.04 and Windows 10 Seeking podcasts related to geospatial technologies? I would like to make a list of podcasts related to geospatial technologies.  Do you listen to any that you find interesting and informative? Layover at Tokyo Narita airport: can I travel outside, and what kind of visa would I need? Django: AppRegistryNotReady() Function returns None without return statement Where to ask "subjective and argumentative" questions on programming? Can “whose” refer to an inanimate object? We lit a fire whose fuel was old timber wood.   Is the word whose referring to fire, an inanimate object, correct in this sentence? Or is there a more appropriate word? Not a problem, but have been seeing questions recently of people not being offered upgrade from 18.04 to 20.04.  I thought these upgrades were offered back in June.  I upgraded to 20.04 from 19.10 and this was offered back in May.  Imagine that different versions paths have different schedules, but google does not help explaining(that I found).  Does someone know the schedules? Invariance, covariance and symmetry Though often heard, often read, often felt being overused, I wonder what are the precise definitions of invariance and covariance. Could you please give me an example from field theory? How do I disable auto-maximizing of newly launched windows in GNOME? Evaluate forecasting ability of GARCH models with RMSE and MAE Motivation for spectral graph theory. Can you help me resolve this paradox for a geodesic orbit? Does it maximize proper time to sit still rather than orbit for 1 year? I believe I have solved a famous open problem. How do I convince people in the field that I am not a crank? I am designing a poster, which has some geometrical shapes (TikZ paths) surrounded by some text. I would like the text to flow neatly around these shapes. I bet there's some clever TikZ magic that could do this.  EDIT: This is the sort of effect I want, sorry to not be clear:    \documentclass[a4paper]{article} \usepackage{tikz} \begin{document} \begin{tikzpicture} \draw (0,0) to (2,-2) to (2,-4) to (4,-4) to [out=up, in=down] (6.5,0); \node [text width=6cm, anchor=north west] at (0,0) { \hspace{5pt}  This is the sort of effect I want to \\ \hspace{23pt} achieve, with text automatically \\ \hspace{36pt} constrained by some \\ \hspace{46pt} geometrical form. Here \\ \hspace{55pt} it's just a curve, \\ \hspace{57pt} but this could \\ \hspace{57pt} also be the \\ \hspace{57pt} text label \\ \hspace{57pt} of a node.}; \end{tikzpicture} \end{document} I know someone who just got an Atrix 4G and previously had an iPhone 3GS. I was asked how he could move his contacts from the iPhone/iTunes onto the Android device. I have never owned an iPhone and have had limited interaction with them. Does anyone know the simplest method of getting iPhone contacts onto an Android device?