boloman0z

2022-06-25

Taylor Series for $\mathrm{log}(x)$

Does anyone know a closed form expression for the Taylor series of the function $f(x)=\mathrm{log}(x)$ where $\mathrm{log}(x)$ denotes the natural logarithm function?

Does anyone know a closed form expression for the Taylor series of the function $f(x)=\mathrm{log}(x)$ where $\mathrm{log}(x)$ denotes the natural logarithm function?

Xzavier Shelton

Beginner2022-06-26Added 26 answers

$-\mathrm{log}(1-x)=x+\frac{{x}^{2}}{2}+\frac{{x}^{3}}{3}+\dots \phantom{\rule{2em}{0ex}}(|x|<1)$

There is no expansion around $x=1$ because the log is singular at $0$.

There is no expansion around $x=1$ because the log is singular at $0$.

Craig Mendoza

Beginner2022-06-27Added 6 answers

For $x\in \mathbb{R}$ satisfying $0<x<2$

$f(x)=\mathrm{ln}(x)=(x-1)-\frac{1}{2}{(x-1)}^{2}+\frac{1}{3}{(x-1)}^{3}-\frac{1}{4}{(x-1)}^{4}+\cdots $

$f(x)={\displaystyle \sum _{n=1}^{\mathrm{\infty}}\left[\frac{{(-1)}^{n+1}}{n}{(x-1)}^{n}\right]}$

$f(x)=\mathrm{ln}(x)=(x-1)-\frac{1}{2}{(x-1)}^{2}+\frac{1}{3}{(x-1)}^{3}-\frac{1}{4}{(x-1)}^{4}+\cdots $

$f(x)={\displaystyle \sum _{n=1}^{\mathrm{\infty}}\left[\frac{{(-1)}^{n+1}}{n}{(x-1)}^{n}\right]}$

$\frac{20b}{{\left(4{b}^{3}\right)}^{3}}$

Which operation could we perform in order to find the number of milliseconds in a year??

$60\cdot 60\cdot 24\cdot 7\cdot 365$ $1000\cdot 60\cdot 60\cdot 24\cdot 365$ $24\cdot 60\cdot 100\cdot 7\cdot 52$ $1000\cdot 60\cdot 24\cdot 7\cdot 52?$ Tell about the meaning of Sxx and Sxy in simple linear regression,, especially the meaning of those formulas

Is the number 7356 divisible by 12? Also find the remainder.

A) No

B) 0

C) Yes

D) 6What is a positive integer?

Determine the value of k if the remainder is 3 given $({x}^{3}+k{x}^{2}+x+5)\xf7(x+2)$

Is $41$ a prime number?

What is the square root of $98$?

Is the sum of two prime numbers is always even?

149600000000 is equal to

A)$1.496\times {10}^{11}$

B)$1.496\times {10}^{10}$

C)$1.496\times {10}^{12}$

D)$1.496\times {10}^{8}$Find the value of$\mathrm{log}1$ to the base $3$ ?

What is the square root of 3 divided by 2 .

write $\sqrt[5]{{\left(7x\right)}^{4}}$ as an equivalent expression using a fractional exponent.

simplify $\sqrt{125n}$

What is the square root of $\frac{144}{169}$