Taylor Series — Question 4

*Question 4 Derive the Maclaurin series for $\log(1+x)$ by integrating a geometric series. Give its first five nonzero terms and determine its exact interval of convergence.

*Question 4 – Solution

For $|t|<1$, $$\frac1{1+t}=\sum_{n=0}^{\infty}(-1)^n t^n.$$

$$\log(1+x)=\sum_{n=0}^{\infty}\frac{(-1)^n x^{n+1}}{n+1}=\boxed{\sum_{n=1}^{\infty}\frac{(-1)^{n-1}x^n}{n}}.$$ The first five nonzero terms are $$x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\frac{x^5}{5}.$$

The interior radius is $1$. At $x=1$, the alternating harmonic series converges to $\log2$. At $x=-1$, the series becomes $-\sum1/n$ and diverges.

The interval is $\boxed{(-1,1]}$.