Question

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n-9n2 1. Consider an = 1+ 2n - 5n2 (a) (3 points) Does the sequence {an} converge or diverge? Show your work. (b) (3 points) Does the series an converge or diverge? Why?

Answer 1

09an = n g n2 = n-gna - tan-on? limana lim n-gn² non no 142n-sn² noo 1+2 - lim = 0-9 040-5 ola seg. Con - this seq. converges to go ① because no žan diverges because lim anto if limanto , then E an diverges 9 È n-3 compare it with Enam no - 2 bn na lim - lim 2 meint. n9777 n>% no on b e linglim manx ma = = lim n²-3n - lim n lim (1 - 2 / 2 n - 31 = mo na no Jn4 Intl Jit to that - 10 ih l is finite a positive then by limit comparison text either both series converge or both series diverge - I diverges hence E n-3 also diverges. Jnttnt1 © Eccos(n)- cosenti) - Sn = Ecos Ck) - Cocktv) = (Cos I - cos2)+(652 -cosz) 4 - -/+ (cosnos-coontly) Sn= cosi cos(n+1) - s= ling sn = lim cool-countU a " cosinthe [=1,] - no & 800 or lies finite) so this series Comerges, diverges because lim Sn does not exist as it is ascillating noo between cosl-) to cost+I. Кс