Determine whether each of the following sequences with the given n-th term is convergent or divergent. Find the limit of those sequences that converge.1) an=sin−1(12cos⁡1n)2) an=2n+2+53n−13) an=(1+n)12n

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Ruben Gibson · Accepted Answer

1)&amp;nbsp;an=sin−1(12cos⁡1n)∑n=1∞sin−1(12cos⁡1n)Apply series divergence testlimn→∞an≠0,&amp;nbsp;∑n=1∞an⇒divergence=limn→∞sin−1(12cos⁡1n)=12limn→∞sin−1(cos⁡1n)Apply limit=sin−1(12cos⁡1∞)=sin−1(12cos⁡(0))&amp;nbsp;[cos⁡(0)=1]=sin−1(12)=π4≠0Thus,&amp;nbsp;an&amp;nbsp;diverges2)&amp;nbsp;an=2n+2+53n−1Now,&amp;nbsp;∑n=0∞2n+2+53n−1=∑n=0∞(2n+23n−1+53n−1)=∑n=0∞(2n+23n−1+∑n=0∞53n−1)Apply ratio testan=2n+23n−1,&amp;nbsp;an+1=2n+33nan=53n−1,&amp;nbsp;an+1=53n⇒limn→∞|an+1an|=limn→∞|2n+33n×3n×3−12n×22|=2×13=23&amp;lt;1, convergesSimilarly,limn→∞|53n×3n−15|=13&amp;lt;1, convergesThus, the sequence converges

Malia Booth · Accepted Answer

c) an=(1+n)12n∑n=1∞(1+n)12nApply series divergence testlimn→∞an=limn→∞(1+n)12nApply limit=(1+(∞))1∞=(1+∞)0=1limn→∞(1+n)12n=limn→∞e12nln⁡(1+n)=1As limn→∞an≠0The sequence diverges

Determine whether each of the following sequences with

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