Question

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Answer 1

consider vn=1/n^(1/2)
\[\lim_{n \rightarrow \infty} \frac{ an }{ bn }=\lim_{n \rightarrow \infty}\frac{n ^{\frac{ 1 }{ 2 }}( n+2)^2 }{ n^3 }*\frac{ n ^{\frac{ 1 }{ 2 }} }{ 1 }\]
\[=\lim_{n \rightarrow \infty}\frac{ n ^{\left( \frac{ 1 }{ 2 }+\frac{ 1 }{ 2 } \right)n^2\left( 1+\frac{ 2 }{ n } \right)^{2}} }{ n^3 }\]
\[\lim_{n \rightarrow \infty}\frac{ n^3\left( 1+\frac{ 2 }{ n } \right)^2 }{ n^3 }=1+0=1\]
which is finite,hence they behave alike.
compare bn with\[n ^{\frac{ 1 }{ p }},p=\frac{ 1 }{ 2 }<1~ hence ~ series ?\]

Answer 2

correction\[\frac{ 1 }{ n^p }\]