Sequences: All three sequences converge Series: Sequences and Series

A sequence a n is defined by the height of the n th bar, which is equal to the area of the n th rectangle. The series s n is the Riemann Sum up to n. The series converges if the area is finite as n  Sequences and Series

The integral test f (x) The sequence a n can be drawn so that it lies entirely above the function f or entirely below it. Thus the convergence is the same for the series and the improper integral a1a1 a2a2 a3a3 a4a4

Remainder estimate using the integral f (x) n anan a n+1 anan n+1 where a n+1 (error in estimate of s from adding the first n terms)