Consider the series and its fifth partial sum .

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Consider the series ∑k=1∞(−1)k+1k2\displaystyle \sum_{k=1}^{\infty}\dfrac{(-1)^{k+1}}{k^{2}} and its fifth partial sum S5S_5. (a) Write S5S_5 as an exact rational number. (b) Without evaluating any additional terms, obtain an upper bound for ∣S−S5∣|S-S_5|, where SS denotes the exact value of the series. Express your bound as a simplified fraction. (c) Use your result from part (b) to state an interval of length equal to your bound that is guaranteed to contain the true sum SS. Present the interval in the form [S5−bound, S5+bound][S_5-\text{bound},\,S_5+\text{bound}].

Related theory

Browse the problem library