Quiz 8 — March 16, 2011 – Section 4 – 9:05-9:55 recitation..Consider the series∞∑n=22n2-1.(a) LetM≥2 be some fxed integer. Find a closed ±ormula ±or the partial sumsM=M∑n=22n2-1. (Comment. It is possible to use the technique o± partial ±ractionsto express this series as a telescoping series.)(b) What is the sum o± the series?Answer.Write2n2-1=An-1+Bn+1. Multiply byn2-1 to see that2 =A(n+ 1) +B(n-1).Plug inn= 1 to see that 1 =A. Plug inn=-1 to see thatB=-1. This worksbecasue1n-1+-1n+ 1=n+ 1-(n-1)n2-1=2n2-1.(a) We see thatsM=Msn=22n2-1=Msn=2b1n-1-1n+ 1B=±12-1-12+1²+±13-1-13+1²+±14-1-14+1
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