# The following advanced exercise use a generalized ratio test to determine convergence of some series that arise in particular applications

ddaeeric 2021-09-30 Answered

The following advanced exercise use a generalized ratio test to determine convergence of some series that arise in particular applications, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if \$ $\underset{n\to \mathrm{\infty }}{lim}\frac{{a}_{2n}}{{a}_{n}}<1/2$ then $\sum {a}_{n}$ converges, while if $\underset{n\to \mathrm{\infty }}{lim}\frac{{a}_{2n+1}}{{a}_{n}}>1/2$ then $\sum {a}_{n}$  diverges. Let ${a}_{n}=\frac{{n}^{\mathrm{ln}n}}{{\left(\mathrm{ln}n\right)}^{n}}$. Show that $\frac{{a}_{2n}}{{a}_{n}}\to 0$ as $n\to \mathrm{\infty }$

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## Expert Answer

Jayden-James Duffy
Answered 2021-10-01 Author has 91 answers
Proved, $\underset{n\to \mathrm{\infty }}{lim}\frac{{a}_{2n}}{{a}_{n}}=0$
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