Hi, I only need help on 2c. Thank you! Vn2 +1Prove that the sequence {an}n=1converges by following the following sequence of steps.nn=1(a) Prove that {a;}, is bounded.n=1(b) Prove that {a%}=1 is decreasing.(c) In a previous HW assignment we showed that if 0 < a < b, then 0 < Va < vb. Use this result tocomplete the proof.

Name: Hi, I only need help on 2c. Thank you! Vn2 +1Prove that the sequence {
Uploaded: 2024-03-20T07:07:37.000Z
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Description: Hi, I only need help on 2c. Thank you! Vn2 +1Prove that the sequence {an}n=1converges by following the following sequence of steps.nn=1(a) Prove that {a;}, is bounded.n=1(b) Prove that {a%}=1 is decreasing.(c) In a previous HW assignment we showed th

Result: If 0<a<b, then 0<a <b The given sequence is: ann=1∞=n2+1nn=1∞ For every n, an>0. From the previous results, (a) an2n=1∞ is bounded. (b) an2n=1∞ is decreasing. Since an>0, an2>0 for every n.Definition: A sequence an is said to be bounded, if there exists a M>0, such that an <M for every n Since an2n=1∞ is bounded, there exists a M>0, such that an2 <M for every n Since 0<an2, 0<an2 <M for every n From the result, for every n 0<an2 <M 0<an <M Hence then sequence an is bounded by M.Definition: A sequence an is said to be decreasing, if for every n, an<an+1 Since an2n=1∞ is decreasing, 0<an2<an+12 From the result, for every n, 0<an2<an+120<an<an+1 Hence, an is decreasing.Result: A bounded increasing (or decreasing ) function converges. an is bounded. an is decreasing Hence, the sequence an converges.The given sequence ann=1∞=n2+1nn=1∞ converges.