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bchow · May 18, 2007, 05:13 PM

hey can someone help me out with this limit

lim (n*19^n) / (20^n)
n->∞

thanks

galactus · May 18, 2007, 05:49 PM

You can use the ratio test to test convergence.

$\frac{(n+1)\cdot{19^{n+1}}}{20^{n+1}}\cdot\frac{20 ^{n}}{n\cdot{19^{n}}}=\frac{19(n+1)}{20n}$

$\lim_{n\to\infty}\frac{19(n+1)}{20n}=\frac{19}{20}$

The limit is < 1, therefore, it converges. But to what? Just look at the denominator, that should tell you something; It's larger than the numerator. Plug in some larger and larger values. What does it approach?

Try rewriting it as $n(\frac{19}{20})^{n}$ .

See what the limit is now?

This problem would be more interesting of you had to find the sum of the series.