Math 1B Section 112 Quiz #6Thursday, 4 October 2007GSI: Theo Johnson-Freydhttp://math.berkeley.edu/∼theojfName:1. True or False (1 pt each) For each of the following statements, decide if it is trueor false. You do not need to show work: I will grade only your answers.(a) If a sequence {an}∞n=1is strictly increasing, and there’s a number M boundingthe sequence from above (i.e. an≤ M for every n), then limn→∞anexists.True This is the (an equivalent) statement of the monotonic sequences theorem.(b) Let f (x) be a function, and define the sequence an= f(n). If limn→∞an= L,then limx→∞f(x) = L.False It’s true that if limx→∞f(x) exists, then this limit equals limn→∞an(which necessarily converges). But just because a sequence converges does notmean that the function converges (e.g. f(x) = sin(πx)).(c) A geometric series converges if and only if the ratio between successive terms ispositive.False The ratio must be strictly more than −1 and strictly less than +1.12. (3 pts) Use the divergence test to show that the following series diverges. (You willneed to actually compute a limit, or explain why the limit is not defined.)∞Xn=1n32n3+ 1We use the divergence test: the limit limn→∞n32n3+1= limn→∞12+−1/42n3+1=12+limn→∞−1/22n3+1=12+0 6= 0. Thus, since the limit is not 0, the series necessarilydiverges.3. (4 pts) Sum the following telescoping series:∞Xn=13(3n − 2)(3n + 1)=34+328+370+ . . .3(3n − 2)(3n + 1)=13n − 2−13n + 1∞Xn=13(3n − 2)(3n + 1)=11−14+14−17+17−110+ . . .=
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