Name:Section: 8:00-9:30 11:00-12:30Math 1B Quiz 5 SOLUTIONSOctober 4, 2007GSI: Rob BayerYou have twenty minutes to complete this quiz. You must show your work.1. (3 pts) Determine whether the sequence an=2 + sin(n)n2+ ln nconverges or diverges. If itconverges, find its value.Since −1 ≤ sin n ≤ 1, we have1n2+ln n≤2+sin nn2+ln n≤3n2+ln n. Both the left and righthand side of this inequality go to 0 as n → ∞, so by the squeeze theorem, the originalsequence must also.2. (4 pts) Determine whether the series∞Xn=22n2− 1converges or diverges. If it converges,find its value.We start with partial fractions:2n2−1=An−1+Bn+1, so 2 = A(n + 1) + B(n − 1) whichgives the equations A + B = 0, A − B = 2. This has A = 1, B = −1, for a solution, so2n2−1=1n−1−1n+1.Let’s try a few partial sums:s2=11−13s3= 1 −13+12−14s4= 1 −13+12−14+13−15= 1 +12−14−15...sn= 1 +12−1n−1n + 1So limn→∞sn=32and the sum is convergent and has value323. (3 pts) True/False. For those that are true, give a (short) explanation of why and forthose that are false, give a counterexample.(a) If anand bnare sequences such that anand an+ bnboth converge, then bnconverges.True If anand an+ bnare both convergent, then (an+ bn) − anis convergent,but this is just bn(b) Every bounded sequence converges.False an= (−1)nis one possible counterexample.(c) If snis the sequence of partial sums forP∞n=1anand limn→∞sn= 0, thenP∞n=1anconverges.True This is actually the definition ofPan=
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