Consider the sequence {An}, whose nth term is given by An=(1+2n)3n Use limits to analyticlly show how we can determine whether the sequence is convergent or divergent. If it is convergent, find the limit of the sequence. Write a sentence to summariae your findings.

Question

Consider the sequence {An}, whose nth term is given by An=(1+2n)3n  Use limits to analyticlly show how we can determine whether the sequence is convergent or divergent. If it is convergent, find the limit of the sequence. Write a sentence to summariae your findings.

Melinda McCombs · Accepted Answer

Step 1  If the limit of the sequence is finite then the series is convergent otherwise divergent.  limn⇒∞(An)=L=limn⇒∞(1+2n)3n  taking log both sides  log⁡(L)=limn⇒∞[log⁡((1+2n)3n)]  log⁡(L)=limn⇒∞[3nlog⁡(1+2n)]  с=limn⇒∞[3n(2n−(2n)22+(2n)33−сs˙)]  сlimn⇒∞[6−2n2+83n3−сs˙]  log⁡(L)=6  L=e6  thus, limn⇒∞(An)=L=e6 (finite)  Thus, the limit of the sequence is finite and hence the given sequence is convergent.

Thomas White · Accepted Answer

Step 1  To find the first four terms, we substitute n=1, 2, 3, 4 in the formula for the nth term  an=12n+1 nth term  a1=12(1)+1=13 n=1  a2=12(2)+1=15 n=2  a3=12(3)+1=17 n=3  a4=12(4)+1=19 n=4  Step 2  Tofind the 100th term, we substitute n=100  a100=12(100)+1=1201 n=100

Consider the sequence \{A_{n}\}, whose nth term is given by

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