Quiz 7 Solution - Solutions to Quiz 7 Determine whether the series converges or diverges 1 n tan 1 n=1 Solution The series diverges Use the Limit

Quiz 7 Solution - Solutions to Quiz 7 Determine whether the...

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Solutions to Quiz 7 Determine whether the series converges or diverges.1.Xn=1tan1n 7The seriesn=1875·(57)n<because it is a convergent geometric series. Hence theclaim follows.3.Xn=1n3e-n4 f(x) = 3x2e-x4+x3e-x4(-4x3) = 3x2e-x4(1-43x4)<0 forx >1,sofis decreasing on (1,). We haveZ1x3e-x4dx= limT→∞ZT1x3e-x4dx= limT→∞ZT4114e-udu= limT→∞h-14e-T4+14e-1i=14e,thus the improper integralR1x3e-x4dx <.
4.Xn=1(-1)nn2n+ 3 Solution: This series converges. We use the Alternating Series Test. Clearly, the seriesis alternating, and we have limn→∞n2n+3= 0. Consider the functionf(x) =x2x+3forx1.We havef(x) =2x+32x-2x(2x+ 3)2=3-2x2x(2x+ 3)2<forx >32. Consequently, the functionfis decreasing on (3/2,). The AlternatingSeries Test now implies the claim.

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