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.

Famous quotes containing the word proof:

“If we view our children as stupid, naughty, disturbed, or guilty of their misdeeds, they will learn to behold themselves as foolish, faulty, or shameful specimens of humanity. They will regard us as judges from whom they wish to hide, and they will interpret everything we say as further proof of their unworthiness. If we view them as innocent, or at least merely ignorant, they will gain understanding from their experiences, and they will continue to regard us as wise partners.”
—Polly Berrien Berends (20th century)

“If any proof were needed of the progress of the cause for which I have worked, it is here tonight. The presence on the stage of these college women, and in the audience of all those college girls who will some day be the nation’s greatest strength, will tell their own story to the world.”
—Susan B. Anthony (1820–1906)

“It comes to pass oft that a terrible oath, with a swaggering accent sharply twanged off, gives manhood more approbation than ever proof itself would have earned him.”
—William Shakespeare (1564–1616)