<s>[INST] Integral with series 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 $10^{-5}$. 
 [/INST] \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.
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