Question

Find whether the following series is convergent or divergent:
${(\frac{2^2}{1^2}-\frac{2}{1})}^{-1}+{(\frac{3^3}{2^3}-\frac{3}{2})}^{-2}+{(\frac{4^4}{3^4}-\frac{4}{3})}^{-3}+....$

Answer 1

$T_n={[{\left(\frac{n+1}{n}\right)}^{n+1}-\frac{n+1}{n}]}^{-n}$

$⟹T_n^{1/n}={[{\left(\frac{n+1}{n}\right)}^{n+1}-\frac{n+1}{n}]}^{-1}$

$⟹T_n^{1/n}={(1+\frac{1}{n})}^{-1}{[{(1+\frac{1}{n})}^n-1]}^{-1}$

$\underset{n\to \infty }{lim}{T}_n^{1/n}=1\cdot (e-1{)}^{-1}[\because \underset{n\to \infty }{lim}{(1+\frac{1}{n})}^n=e]$

$\therefore \underset{n\to \infty }{lim}{T}_n^{1/n}=\frac{1}{e-1}<1$

Therefore according the Cauchy's test, this series is convergent