Consider a sequence {an}nen such that an 0, {an}neN is decreasing and lim an = 0....

 

Transcribed Image Text:

Consider a sequence {an}nen such that an 0, {an}neN is decreasing and lim an = 0. Consider sn = a1 a2+ a3 22 (-1) k-1 n-x Show that the sequence {s2n-1}nEN = (81, 83, 85,...) is decreasing. Show that the sequence {s2}EN (82, 84, 86...) is increasing. = - +(-1)-an Show that 82n-1 28 for any k > 2n-1 (any odd term is an upper bound for all terms after that). Show that 82 sk for all k >2n (any even term is a lower bound for all term after that). Use the results above to show that {82n-1}neN and (82n-1}EN are convergent. Prove that they have the same limit. = Show that the series 1 - +13 1+. converges. Consider a sequence {an}nen such that an 0, {an}neN is decreasing and lim an = 0. Consider sn = a1 a2+ a3 22 (-1) k-1 n-x Show that the sequence {s2n-1}nEN = (81, 83, 85,...) is decreasing. Show that the sequence {s2}EN (82, 84, 86...) is increasing. = - +(-1)-an Show that 82n-1 28 for any k > 2n-1 (any odd term is an upper bound for all terms after that). Show that 82 sk for all k >2n (any even term is a lower bound for all term after that). Use the results above to show that {82n-1}neN and (82n-1}EN are convergent. Prove that they have the same limit. = Show that the series 1 - +13 1+. converges.