Determine the smallest positive integer n such that the remainder satisfies the inequality $\displaystyle R_n<0.005$.
1. $\displaystyle \sum_{i=1}^{\infty} \frac {1}{n^3+1}$
2. $\displaystyle \sum_{i=1}^{\infty} \frac {cos^4{n}}{n^4}$
Determine the smallest positive integer n such that the remainder satisfies the inequality $\displaystyle R_n<0.005$.
1. $\displaystyle \sum_{i=1}^{\infty} \frac {1}{n^3+1}$
2. $\displaystyle \sum_{i=1}^{\infty} \frac {cos^4{n}}{n^4}$