Wednesday, September 28, 2022
HomeMathRatio take a look at counterexample by Cesàro

Ratio take a look at counterexample by Cesàro




Given a sequence a1, a2, a3, … let L be the restrict of the ratio of consecutive phrases:

Then the sequence

sum_{n=1}^infty a_n

converges if L < 1 and diverges if L > 1.

Nonetheless, that’s not the total story. Right here is an instance from Ernesto Cesàro (1859–1906) that reveals the ratio take a look at to be extra delicate than it could appear at first. Let 1 < α < β and take into account the sequence

frac{1}{1^alpha} + frac{1}{2^beta} + frac{1}{3^alpha} + frac{1}{4^beta} + cdots

The ratio a2n + 1 / a2n diverges, however the sum converges.

Our assertion of the ratio take a look at above is incomplete. It ought to say if the restrict exists and equals L, then the sequence converges if L < 1 and diverges if L > 1. The take a look at is inconclusive if the restrict doesn’t exist, as in Cesàro’s instance. It’s additionally inconclusive if the restrict exists however equals 1.

Cesàro’s instance interweaves two convergent sequence, one consisting of the even phrases and one consisting of the odd phrases. Each converge, however the sequence of even phrases converges quicker as a result of β > α.

Associated publish: Cesàro summation





RELATED ARTICLES

LEAVE A REPLY

Please enter your comment!
Please enter your name here

Most Popular

Recent Comments