Integral Test and P-Series

Introduction

If we add an infinite series of numbers will the sum be infinite or a finite number ?. The series is said to diverge if the answer is infinite, and series is said to converge if the sum has a finite value such as a:

A student first encountering this concept can be forgiven for thinking that the sum of an infinite series is always infinite, since there are an infinite number of terms; however this is not true for an infinite geometric series which has the form:

The sum of a geometric series is convergent provided ∣ r ∣ <1 where r is the common ratio. The latter restriction on r ensure that the next term in the series is small that previous term i.e.

this however is not the condition for the convergence of all series. All decreasing series do not converges we study and compare the harmonic series which is the sum of the reciprical of positive integers

with the infinite sum of the reciprocal of the square numbers. Both series we shall see later are special cases of the p-series.

Calculating the Partial Sums

In the figure below the partial sums of the two series are calculated. When k=1 both series equal 1. The value of k can be increased or decreased using the buttons.