integral test for convergence

3 min read 19-03-2025

The question of whether an infinite series converges or diverges is a fundamental one in calculus. Many techniques exist to answer this, and the Integral Test provides a powerful method for dealing with series whose terms are positive and decreasing. This article will explore the Integral Test for convergence, providing examples and explanations to solidify your understanding.

Understanding the Integral Test

The Integral Test for convergence establishes a connection between the convergence of an infinite series and the convergence of an improper integral. Specifically, it states:

Theorem: Let ∑_n=1^∞ f(n) be a series where f(x) is a continuous, positive, and decreasing function for x ≥ 1. Then the series ∑_n=1^∞ f(n) converges if and only if the improper integral ∫₁^∞ f(x) dx converges.

This theorem provides a powerful tool. Instead of directly analyzing the series' sum, we can analyze a corresponding integral, which can often be easier to evaluate.

Conditions for the Integral Test

It's crucial to remember that the Integral Test only applies when three conditions are met:

Continuous: The function f(x) must be continuous for x ≥ 1. This means there are no breaks or jumps in the function's graph within this interval.
Positive: The function f(x) must be positive for x ≥ 1. This means f(x) > 0 for all x ≥ 1.
Decreasing: The function f(x) must be decreasing for x ≥ 1. This means that f(x) ≥ f(x+1) for all x ≥ 1.

Applying the Integral Test: Examples

Let's illustrate the Integral Test with a few examples:

Example 1: The Harmonic Series

Consider the harmonic series: ∑_n=1^∞ (1/n). Let f(x) = 1/x. This function is continuous, positive, and decreasing for x ≥ 1.

Now, let's evaluate the corresponding improper integral:

∫₁^∞ (1/x) dx = lim_t→∞ [ln|x|]₁^t = lim_t→∞ (ln|t| - ln|1|) = ∞

Since the integral diverges, the harmonic series also diverges according to the Integral Test.

Example 2: A Convergent p-series

Let's examine the p-series ∑_n=1^∞ (1/n²). Here, f(x) = 1/x². Again, this function is continuous, positive, and decreasing for x ≥ 1.

The corresponding integral is:

∫₁^∞ (1/x²) dx = lim_t→∞ [-1/x]₁^t = lim_t→∞ (-1/t + 1) = 1

Since the integral converges to 1, the p-series ∑_n=1^∞ (1/n²) also converges by the Integral Test. This highlights that p-series converge when p > 1.

Example 3: A More Complex Case

Consider the series ∑_n=2^∞ 1/(nln(n)). Let f(x) = 1/(xln(x)). For x ≥ 2, this function is continuous, positive, and decreasing.

The corresponding integral is:

∫₂^∞ 1/(x*ln(x)) dx = lim_t→∞ [ln|ln|x||]₂^t = ∞

Since the integral diverges, the series ∑_n=2^∞ 1/(n*ln(n)) also diverges.

When the Integral Test is Useful

The Integral Test shines when dealing with series where the terms can be expressed as a function that is easily integrable. It's especially helpful for series that resemble p-series or other easily integrable functions. However, remember that it only applies to series with positive, decreasing, and continuous terms. For other types of series, different convergence tests might be more appropriate.

Conclusion

The Integral Test offers a valuable approach to determining the convergence or divergence of infinite series. By connecting series to improper integrals, it provides a straightforward method for analyzing the behavior of certain series, particularly those with easily integrable terms satisfying the necessary conditions. Understanding its application and limitations is crucial for mastering the diverse landscape of convergence tests.