P Series Test

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:

Where p is a positive real number.

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:

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:

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: converges to 2.

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

• For example, the harmonic series: diverges, meaning it grows without bound as more terms are added.
• The harmonic series is the sum of the reciprocals of the positive integers.

The p-series test can be used to determine the convergence of .
According to the p-series test:

• ​ will converge when
  When p is greater than 1, the terms decrease sufficiently fast as n increases, leading the series to sum to a finite value.

• ​ will diverge when
  When p is less than or equal to 1, the terms do not decrease quickly enough to prevent the series from growing without bound, resulting in divergence.

Examples of P-Series

• Convergent p-Series:
  • . This series converges.
    
    
• Divergent p-Series:
  • p = 1 (Harmonic Series): . This series diverges.
    
    
• Divergent p-Series:
  • p = 1/2, . 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: 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 p ≤ 1, the series diverges.

Let's consider examples for better understanding:

Example 1: Consider the series:

, find out it is convergent or divergent?

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:

, find out it is convergent or divergent?

Solution:

Identify the Series: This can be written as:
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:

, find out it is convergent or divergent?

Solution:

Simplify the general term: This can be written as:

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:

Read More,

• Conditional Convergence
• Absolute and Conditional Convergence
• Radios of Convergence

Practice Problems on P Series Test

Problem 1: Determine the convergence or divergence of the series:

Problem 2: Determine the convergence or divergence of the series:

Problem 3: Determine the convergence or divergence of the series:

Problem 4: Determine the convergence or divergence of the series:

Problem 5: Determine the convergence or divergence of the series: