New Notes Blog: `a_n = (10n^2+3n+7)/(2n^2-6)` Determine the convergence or divergence of the sequence with the given n'th term. If the sequence converges,...

Friday, 2 October 2015

$\displaystyle{a}_{{n}}=\frac{{{10}{{n}}^{{2}}+{3}{n}+{7}}}{{{2}{{n}}^{{2}}-{6}}}$ Determine the convergence or divergence of the sequence with the given n'th term. If the sequence converges,...

$\displaystyle\lim_{{{n}\to\infty}}\frac{{{10}{{n}}^{{2}}+{3}{n}+{7}}}{{{2}{{n}}^{{2}}-{6}}}=$

Divide both numerator and the denominator by $\displaystyle{{n}}^{{2}}.$

$\displaystyle\lim_{{{n}\to\infty}}\frac{{\frac{{{10}{{n}}^{{2}}}}{{{n}}^{{2}}}+\frac{{{3}{n}}}{{{n}}^{{2}}}+\frac{{7}}{{{n}}^{{2}}}}}{{\frac{{{2}{{n}}^{{2}}}}{{{n}}^{{2}}}-\frac{{6}}{{{n}}^{{2}}}}}=\lim_{{{n}\to\infty}}\frac{{{10}+\frac{{3}}{{n}}+\frac{{7}}{{{n}}^{{2}}}}}{{{2}-\frac{{6}}{{{n}}^{{2}}}}}=$

Since $\displaystyle\lim_{{{n}\to\infty}}\frac{\alpha}{{{n}}^{\beta}}={0},$ $\displaystyle\forall\alpha\in\mathbb{R},$ $\displaystyle\forall\beta>{0}$ we have

$\displaystyle\frac{{{10}+{0}+{0}}}{{{2}-{0}}}=\frac{{10}}{{2}}={5}$

As we can see the sequence converges to 5.

The image below shows first 150 terms of the sequence. We can see they are asymptotically approaching the red line whose equation is $\displaystyle{y}={5}.$

$\displaystyle\lim_{{{n}\to\infty}}\frac{{{10}{{n}}^{{2}}+{3}{n}+{7}}}{{{2}{{n}}^{{2}}-{6}}}=$

Divide both numerator and the denominator by $\displaystyle{{n}}^{{2}}.$

Since $\displaystyle\lim_{{{n}\to\infty}}\frac{\alpha}{{{n}}^{\beta}}={0},$ $\displaystyle\forall\alpha\in\mathbb{R},$ $\displaystyle\forall\beta>{0}$ we have

$\displaystyle\frac{{{10}+{0}+{0}}}{{{2}-{0}}}=\frac{{10}}{{2}}={5}$

As we can see the sequence converges to 5.

The image below shows first 150 terms of the sequence. We can see they are asymptotically approaching the red line whose equation is $\displaystyle{y}={5}.$

New Notes Blog

Friday, 2 October 2015

$\displaystyle{a}_{{n}}=\frac{{{10}{{n}}^{{2}}+{3}{n}+{7}}}{{{2}{{n}}^{{2}}-{6}}}$ Determine the convergence or divergence of the sequence with the given n'th term. If the sequence converges,...

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In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?

Friday, 2 October 2015

Determine the convergence or divergence of the sequence with the given n'th term. If the sequence converges,...

No comments:

Post a Comment

In &quot;By the Waters of Babylon,&quot; under the leadership of John, what do you think the Hill People will do with their society?

$\displaystyle{a}_{{n}}=\frac{{{10}{{n}}^{{2}}+{3}{n}+{7}}}{{{2}{{n}}^{{2}}-{6}}}$ Determine the convergence or divergence of the sequence with the given n'th term. If the sequence converges,...

In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?