Answer Happy

ALTERNATING SERIES TEST: If the series n=1∑∞​(−1)n+1bn​=b1​−b2​+b3​−b4​+⋯ satisfies (i) bn​≥bn+1​ for n≥N (that is, the sequence {bn​} is eventually decreasing), and (ii) n→∞lim​bn​=0, then the series converges. In problems 1 (a-e) use the Alternating Series Test to determine if the series converges. If the Alternating Series Test does not give convergence, apply another test to determine whether the series converges or diverges.
(a) ln21​−ln31​+ln41​−ln51​+⋯ Converges (b) n=1∑∞​n​(−1)n−1​ Converges (c) n=1∑∞​(−1)n2n+13n−1​ Diverges (d) n=1∑∞​4n2+1(−1)n+1​ Converges (e) n=1∑∞​(−1)n+1n3+4n2​ Converges