New Notes Blog: `sum_(n=0)^oo (n+10)/(10n+1)` Determine the convergence or divergence of the series.

Wednesday, 25 June 2014

$\displaystyle{\sum_{{{n}={0}}}^{\infty}}\frac{{{n}+{10}}}{{{10}{n}+{1}}}$

For the series $\displaystyle{a}_{{n}}=\frac{{{n}+{10}}}{{{10}{n}+{1}}}$

$\displaystyle{a}_{{n}}=\frac{{{1}+\frac{{10}}{{n}}}}{{{10}+\frac{{1}}{{n}}}}$

$\displaystyle\lim_{{{n}\to\infty}}{a}_{{n}}=\lim_{{{n}\to\infty}}\frac{{{1}+\frac{{10}}{{n}}}}{{{10}+\frac{{1}}{{n}}}}$

$\displaystyle=\frac{{1}}{{10}}\ne{0}$

As per the n'th term test for divergence,

If $\displaystyle\lim_{{{n}\to\infty}}{a}_{{n}}\ne{0}$ , then $\displaystyle{\sum_{{{n}={1}}}^{\infty}}{a}_{{n}}$ diverges

So, the series diverges.

$\displaystyle{\sum_{{{n}={0}}}^{\infty}}\frac{{{n}+{10}}}{{{10}{n}+{1}}}$

For the series $\displaystyle{a}_{{n}}=\frac{{{n}+{10}}}{{{10}{n}+{1}}}$

$\displaystyle{a}_{{n}}=\frac{{{1}+\frac{{10}}{{n}}}}{{{10}+\frac{{1}}{{n}}}}$

$\displaystyle\lim_{{{n}\to\infty}}{a}_{{n}}=\lim_{{{n}\to\infty}}\frac{{{1}+\frac{{10}}{{n}}}}{{{10}+\frac{{1}}{{n}}}}$

$\displaystyle=\frac{{1}}{{10}}\ne{0}$

As per the n'th term test for divergence,

If $\displaystyle\lim_{{{n}\to\infty}}{a}_{{n}}\ne{0}$ , then $\displaystyle{\sum_{{{n}={1}}}^{\infty}}{a}_{{n}}$ diverges

So, the series diverges.

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