Consider the series

MATHEMATICS · Subjective · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Consider the series

T=\displaystyle\sum_{n=1}^{\infty}(-1)^{\,n}\dfrac{1}{n^{2}}

. A student uses the first six terms and obtains

T\approx0.822

. The student asserts, “This value is accurate to three decimal places.” Explain whether the assertion is justified.