Question

How to Expand the following as power series? `e^x` and `ln(1+x)`

Answer 1

Please see below.

Explanation:

A power series is te series that can be written as `sum_(n=0)^(oo)c_nx^n`.

Here it would be better to use a special version of it called Taylor series according to which

`f(x)=sum_(n=0)^(oo)f^n(0)/(n!)x^n`

Note that for `f(x)=e^x`, we have for all `n`, `f^n(x)=e^x` and hence `f^n(0)=1` and

`f(x)=e^x=sum_(n=0)^(oo)x^n/(n!)=1+x/(1!)+x^2/(2!)+x^3/(3!)+......`

For `f(x)=ln(1+x)`, we know `1/(1+x)=1-x+x^2-x^3+.......`

(you just multiply RHS by `1+x` and will get `1` for `x in(-1,1)`)

Integrating `ln(1+x)=x-x^2/2+x^3/3-x^4/4+x^5/5-........`

or `ln(1+x)=sum_(n=0)^(oo)(-1)^nx^(n+1)/(n+1)`

= `sum_(n=1)^(oo)(-1)^(n+1)x^n/n`