Integral with series

Question

How do I represent this integral

$$\int_{0}^{1} \frac{10}{10+x^4} dx$$

as a series so that I can calculate with an error of less than $10^{-5}$.

Answer 1

Hint: $$\frac{1}{1 + \frac{x^4}{10}} = \sum_{n=0}^{\infty}(-1)^n \frac{x^{4n}}{10^n}$$

where $|x| < 1$

Answer 2

\begin{align} \frac{10}{10+x^4} = \frac{1}{1-\left( \frac{-x^4}{10} \right)} = \frac 1 {1-r} & = 1+r+r^2+r^3+\cdots \\[6pt] & = 1-\frac{x^4}{10} + \frac{x^8}{100} - \cdots \end{align} Integrating term by term from $0$ to $1$ gives $$ 1 - \frac 1 {50} + \frac 1 {900} -\cdots $$ Since the terms alternate in sign and get smaller, and approach $0$, the error after each term is always smaller in absolute value than the next term.