P+Series

//P// - Series A generalization of the harmonic series is the ** // p //-series **, defined as:    ﻿ for any **positive** real number //p//. **NOTE: //In order for the P-series test to work, the series MUST be in the form shown above//**

**If 0 < p ≤ 1 then the series diverges. **
If the series fits the p-series form and the condition of p being a positive real number, the test will never be inconclusive. If the series converges and you would like to find the value to which the series converges, you can use the sum of sequence method on your calculator.

Hit the following keys:

1.) 2nd List 2.) Scroll over to Math 3.) Hit sum ( 4.) Hit 2nd list again 5.) Scroll over to OPS <span style="font-family: 'Times New Roman',Times,serif;">6.) Hit seq( 7.) Type in function 8.) Type:, x , 1, 50 (or another large number) 9.) Hit <span style="font-family: 'Times New Roman',Times,serif;">ENTER

<span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">Examples <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"><span style="background-color: transparent; color: #000000; display: block; font-family: 'Times New Roman',Times,serif; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; font-family: 'Times New Roman',Times,serif; font-size: 16px; text-decoration: none; vertical-align: auto;">is a P-series <span style="background-color: #ffffff; color: #000000; font-family: 'Times New Roman',Times,serif; font-size: 16px; vertical-align: auto;">with p = 1/2. Recall that roots are just 1/2 powers and pulling up from the denominator is just changing the sign of the exponent. Because p < 1, the series diverges <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"><span style="background-color: transparent; color: #000000; display: block; font-family: 'Times New Roman',Times,serif; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; font-family: 'Times New Roman',Times,serif; font-size: 16px; text-decoration: none; vertical-align: auto;">is a P-series with p = 1.This is just the //harmonic series//. Because p = 1, the series diverges. <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"><span style="background-color: transparent; color: #000000; display: block; font-family: 'Times New Roman',Times,serif; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; font-family: 'Times New Roman',Times,serif; font-size: 16px; text-decoration: none; vertical-align: auto;">is not a P-series since the index is in the exponent. In fact the above series is a geometric series with r = 1/2.