Mathematics

Problem

The series

Q=\displaystyle\sum_{n=1}^{\infty}(-1)^{\,n+1}\dfrac{1}{n^{5}}

is approximated by the partial sum

Q_{10}

. Determine whether

Q_{10}

is guaranteed to approximate

Q

with absolute error less than

1\times10^{-6}

. Provide a numerical bound to support your answer.