Chapter 11.5, Problem 27E

To determine

The Taylor series of $\ln \left(x+\sqrt{1+x^2}\right)$ at $x=0$, Where the Taylor series expansion of $\frac{1}{\sqrt{1+x}}$ is $1-\frac{1}{2}x+\frac{1\cdot 3}{2\cdot 4}{x}^2-\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}{x}^3+\frac{1\cdot 3\cdot 5\cdot 7}{2\cdot 4\cdot 6\cdot 8}{x}^4-\cdot \cdot \cdot$ and ${\displaystyle \int \frac{1}{\sqrt{1+x^2}}}dx=\ln \left(x+\sqrt{1+x^2}\right)+C$