Math 1B Section 107 Quiz #8Thursday, 18 October 2007GSI: Theo Johnson-Freydhttp://math.berkeley.edu/∼theojfName:For the first two questions, the alternating series diverge. For each series, decide whichparts of the Alternating Series Test it satisfies, and which parts it fails to satisfy:1. (2 pts)∞Xn=1(−1)nn + 12nThe alternating series test demands that a series be(a) alternating: eitherP(−1)nanorP(−1)n−1an, where an≥ 0.(b) decreasing: an≥ an+1(c) tending towards 0: limn→∞an= 0.This series satisfied (a) and (b), but fails to have the correct limit:limn→∞n + 12n=122. (2 pts) 1 −14+13−116+15−164+17−1256+19−11024+111−14096+ . . .This series satisfies parts (a) and (c) above, but fails to satisfy part (b); for example,15>116, and17>164.1For the next two questions, use the Ratio Test to determine if the series converges ordiverges.3. (3 pts)∞Xn=0n!14nWe use the ratio test:limn→∞(n + 1)!14n+1n!14n= limn→∞(n + 1)14= +∞ > 1so the series diverges.4. (3 pts)∞Xn=0n2n+ 1We use the ratio test:limn→∞n+12n+1+1n2n+1= limn→∞n + 1n2n+ 12n+1+ 1= limn→∞1 +12n2 +12n=12< 1so the series converges
View Full Document
Unlocking...