Suppose an alternating series with terms that are nonincreasing in magnitude, converges to S and the sum of the first n terms of the series is Sn. Suppose also that the difference between the magnitudes of consecutive terms decreases with k. It can be shown that for n ≥ 1,

a. Interpret this inequality and explain why it is a better approximation to S than S_n.
b. For the following series, determine how many terms of the series are needed to approximate its exact value with an error less than 10^-6 using both S_n and the method explained in part (a).

(i)

(ii)

(iii) 

Transcribed Image Text:

Σ-1). a. +1 (-1)"* 'an+1 S, + |an+1 – An+2|. VI