Consider the infinite series S= _ n=1 ^ (-1)^ n+1 , 1 n+1

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

Problem

Consider the infinite series

S=\sum_{n=1}^{\infty}(-1)^{n+1}\,\frac{1}{n+1}.

Let

S_5

denote the partial sum obtained by adding the first five terms of the series. (1) Verify that the sequence of term magnitudes

\left\{\dfrac{1}{n+1}\right\}_{n\ge 1}

is positive, eventually decreasing, and converges to

0

. Provide clear reasoning for each of the three properties. (2) Compute the magnitude of the next term that would be added after

S_5

, namely

|a_6|

. (3) Explain what your result from part (b) implies about the largest possible difference between

S

and

S_5

. State your conclusion in one complete sentence that links the calculation to the accuracy of

S_5

as an approximation of

S