Ratio Test - Proof

Proof

Below is a proof of the validity of the original ratio test.

Suppose that . We can then show that the series converges absolutely by showing that its terms will eventually become less than those of a certain convergent geometric series. To do this, let . Then r is strictly between L and 1, and for sufficiently large n (say, n greater than N). Hence for each n > N and i > 0, and so

$\sum_{i=N+1}^{\infty}|a_{i}| = \sum_{i=1}^{\infty}|a_{N+i}| < \sum_{i=1}^{\infty}r^{i}|a_{N+1}| = |a_{N+1}|\sum_{i=1}^{\infty}r^{i} = |a_{N+1}|\frac{r}{1 - r} < \infty.$

That is, the series converges absolutely.

On the other hand, if L > 1, then for sufficiently large n, so that the limit of the summands is non-zero. Hence the series diverges.

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