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2022-03-25

Determine whether the sequences is convergent or divergent
${a}_{n}=\mathrm{ln}\left(\frac{2{n}^{2}-n+1}{2{n}^{2}+10n-7}\right)$
${a}_{n}=\frac{{2}^{n}}{n!}$

aznluck4u72x4

1)${a}_{n}=\mathrm{ln}\left(\frac{2{n}^{2}-n+1}{2{n}^{2}+10n-7}\right)$
$\underset{n\to \mathrm{\infty }}{lim}{a}_{n}=\underset{n\to \mathrm{\infty }}{lim}\mathrm{ln}\left(\frac{2{n}^{2}-n+1}{2{n}^{2}+10n-7}\right)$
$=\underset{n\to \mathrm{\infty }}{lim}\mathrm{ln}\left(\frac{{n}^{2}\left(2-\frac{1}{n}+\frac{1}{{n}^{2}}\right)}{{n}^{2}\left(2+\frac{10}{n}-\frac{7}{{n}^{2}}\right)}\right)$
$=\underset{n\to \mathrm{\infty }}{lim}\mathrm{ln}\left(\frac{2-\frac{1}{n}+\frac{1}{{n}^{2}}}{2+\frac{10}{n}-\frac{7}{{n}^{2}}}\right)$
Substitute $n=\mathrm{\infty }$
$=\mathrm{ln}\left(\frac{2-\frac{1}{\mathrm{\infty }}+\frac{1}{\mathrm{\infty }}}{2+\frac{10}{\mathrm{\infty }}-\frac{7}{\mathrm{\infty }}}\right)$
$=\mathrm{ln}\left(\frac{2}{2}\right)=\mathrm{ln}1=0$
Thus, the sequence converges

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