MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
Consider the two infinite series below.
(i) n=1∑∞n(−1)n+1
(ii) n=1∑∞n21
Answer the following.
(1) For series (i), justify that the series converges but does not converge absolutely.
(2) For series (ii), justify that the series converges absolutely.
(3) Describe, in complete sentences, the quantitative difference between absolute convergence and conditional convergence illustrated by these two examples.
(4) Explain why any rearrangement of the terms of series (ii) must have the same numerical sum, whereas a rearrangement of the terms of series (i) can alter its sum.