p-series test is a fundamental tool in mathematical analysis used to determine the convergence or divergence of a specific type of infinite series known as p-series. A p-series is defined by the general form:

[Tex]\sum_{n=1}^{\infty} \frac{1}{n^p}[/Tex]

Where p is a positive real number. The p-series test provides a simple criterion to decide the behavior of such series. This test is particularly useful because it allows for quick assessments of series convergence, which is essential in many areas of calculus, number theory, and applied mathematics.

What is a Series in Mathematics?

In mathematics, a series is the sum of the terms of a sequence. Given a sequence of numbers a₁, a₂, a₃, . . ., a series is the expression formed by adding these numbers together. For example, the series corresponding to the sequence a₁, a₂, a₃, . . ., is written as:

S = a₁ + a₂ + a₃ + . . .

Finite Series: If the sequence has a finite number of terms, the series is finite. For example, the sum of the first n natural numbers:

S_n = 1 + 2 + 3 + ⋯ + n

Infinite Series: If the sequence has an infinite number of terms, the series is infinite. For example, the sum of the reciprocals of the natural numbers:

S = 1 + 1/2 + 1/3 + 1/4 + ⋯

Convergence and Divergence of Series

Convergent Series: An infinite series is said to converge if the sum of its terms approaches a finite number as more terms are added.

• For example, the geometric series: S =1 + 1/2 + 1/4 + 1/8 + . . . converges to 2.

Divergent Series: If the sum does not approach a finite limit, the series is divergent.

• For example, the harmonic series: S = 1 + 1/2 + 1/3 + 1/4 + ⋯ diverges, meaning it grows without bound as more terms are added.

What is the P Series Test?

p-series test is a method used to determine the convergence or divergence of a specific type of infinite series known as the ppp-series. A ppp-series is a series of the form:

[Tex]\sum_{n=1}^{\infty} \frac{1}{n^p}[/Tex]

Where p is a positive real number. The convergence of the p-series depends on the value of p.

Conditions for Convergence

The series [Tex]\sum_{n=1}^{\infty} \frac{1}{n^p}​[/Tex] converges if p > 1.

Explanation: Converges for p > 1: When p is greater than 1, the terms \frac{1}{n^p} decrease sufficiently fast as n increases, leading the series to sum to a finite value.

Conditions for Divergence

The series [Tex]\sum_{n=1}^{\infty} \frac{1}{n^p}​[/Tex] diverges if 0 < p ≤ 1.

Explanation: Diverges for 0 < p ≤ 1: When p is less than or equal to 1, the terms 1/n^p do not decrease quickly enough to prevent the series from growing without bound, resulting in divergence.

Examples of P-Series

• Convergent p-Series:
  • [Tex]\sum_{n=1}^{\infty} \frac{1}{n^2} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots[/Tex]. This series converges.
• Divergent p-Series:
  • p = 1 (Harmonic Series): [Tex]\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots [/Tex]. This series diverges.
• Divergent p-Series:
  • p = 1/2, [Tex]\sum_{n=1}^{\infty} \frac{1}{n^{1/2}} = 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{4}} + \cdots[/Tex]. This series diverges.

How to Apply the P Series Test?

We can use the following steps, to apply the p series test to any appropriate series:

Step 1: Identify the Series.

Determine if the series in question can be written in the form: [Tex]\sum_{n=1}^{\infty} \frac{1}{n^p}​[/Tex]where p is a positive real number.

Step 2: Compare with the p-Series.

Check if the given series matches the standard p-series format or if it can be compared to it.

Step 3: Determine the Value of p.

Apply the Test:

• If p > 1, the series converges.
• If 0 < p ≤ 1, the series diverges.

Let’s consider examples for better understanding:

Example 1: Consider the series:

[Tex]\sum_{n=1}^{\infty} \frac{1}{n^3}[/Tex]

Solution:

1. Identify the Series: This is a standard p-series with p = 3.
2. Determine the Value of p: Here, p = 3.
3. Apply the Test: Since p = 3 > 1, the series converges.

Example 2: Consider the series:

[Tex]\sum_{n=1}^{\infty} \frac{2}{n^{1.5}}[/Tex]

Solution:

Identify the Series: This can be written as: [Tex]\sum_{n=1}^{\infty} \frac{2}{n^{1.5}} = 2 \sum_{n=1}^{\infty} \frac{1}{n^{1.5}}[/Tex]

Determine the Value of p: Here, p = 1.5.

Apply the Test: Since p = 1.5 > 1, the series converges.

Example 3: Consider the series:

[Tex]\sum_{n=1}^{\infty} \frac{3}{(2n)^2}[/Tex]

Solution:

Simplify the general term: [Tex]\frac{3}{(2n)^2} = \frac{3}{4n^2} = \frac{3}{4} \cdot \frac{1}{n^2}(2n)[/Tex]

This can be written as: [Tex]\frac{3}{4} \sum_{n=1}^{\infty} \frac{1}{n^2}[/Tex]

Determine the Value of p: Here, p = 2

Apply the Test: Since p = 2 > 1, the series converges. The factor 3/4​ does not affect convergence.

P Series Vs Ratio Vs Root Test

The key differences between p-series, ratio and root test are listed in the following table: