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:

a_n=\sum_{n=1}^{\infty} \frac{1}{n^p} = \frac {1} {1^p} + \frac {1} {2^p} + \frac {1} {3^p}+ \frac {1} {4^p} +⋯

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:

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 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots

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 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots 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 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots 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 a_n.
According to the p-series test:

a_n will converge when 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.
a_n will diverge when p \leq 1
When p is less than or equal to 1, the terms \frac{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:
- \sum_{n=1}^{\infty} \frac{1}{n^2} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots. This series converges.
Divergent p-Series:
- p = 1 (Harmonic Series): \sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots . This series diverges.
Divergent p-Series:
- p = 1/2, \sum_{n=1}^{\infty} \frac{1}{n^{1/2}} = 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{4}} + \cdots. 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: \sum_{n=1}^{\infty} \frac{1}{n^p}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:

\sum_{n=1}^{\infty} \frac{1}{n^3}, find out it is convergent or divergent?

Solution:

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

Example 2: Consider the series:

\sum_{n=1}^{\infty} \frac{2}{n^{1.5}}, find out it is convergent or divergent?

Solution:

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

\sum_{n=1}^{\infty} \frac{3}{(2n)^2}, find out it is convergent or divergent?

Solution:

Simplify the general term: \frac{3}{(2n)^2} = \frac{3}{4n^2} = \frac{3}{4} \cdot \frac{1}{n^2}(2n)This can be written as: \frac{3}{4} \sum_{n=1}^{\infty} \frac{1}{n^2}

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:

Aspect	P-Series Test	Ratio Test	Root Test
Definition	Tests the convergence of series of the form \frac{1}{n^p}	Tests the convergence based on the ratio of successive terms	Tests the convergence based on the n^th root of terms
Formula	\sum_{n=1}^{\infty} \frac{1}{n^p}	L = \lim_{n \to \infty} \frac{a_{n+1}}{a_n}	L =\lim_{n \to \infty } \sqrt[n]{\| {a_n}\|} = \lim_{n \to \infty } {\| a_n\|}^{\frac{1}{n}}
Convergence Criteria	Converges if p > 1 Diverges if p ≤ 1	Converges if L < 1 Diverges if L > 1 Inconclusive if L = 1	Converges if L < 1 Diverges if L > 1 Inconclusive if L = 1
Applicability	Only for series of the form \sum\frac{1}{n^p}	General series ∑a_n	General series ∑a_n
Use Cases	Useful for harmonic series and similar forms	Effective for series with factorials or exponential terms	Effective for series with terms raised to n^th power
Example Series	∑1/n²	∑n!/2n	∑(1/n)ⁿ
Ease of Use	Simple to apply for specific form	Requires computation of limit of ratio	Requires computation of limit of n^th root
Convergence Test Type	Special case test	Ratio-based convergence test	Root-based convergence test

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Practice Problems on P Series Test

Problem 1: Determine the convergence or divergence of the series: \sum_{n=1}^{\infty} \frac{1}{n^4}

Problem 2: Determine the convergence or divergence of the series: \sum_{n=1}^{\infty} \frac{1}{n^{0.5}}

Problem 3: Determine the convergence or divergence of the series: \sum_{n=1}^{\infty} \frac{1}{n^3}

Problem 4: Determine the convergence or divergence of the series: \sum_{n=1}^{\infty} \frac{1}{n}

Problem 5: Determine the convergence or divergence of the series: \sum_{n=1}^{\infty} \frac{1}{n^{1.2}}

Convergence and Divergence of Series

Examples of P-Series

How to Apply the P Series Test?

P Series Vs Ratio Vs Root Test

Practice Problems on P Series Test

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