The Integral Test

The Integral Test is a method used to determine whether an infinite series converges or diverges. It is particularly useful for series whose terms resemble the values of a function that is continuous, positive, and decreasing for $x \geq N$ (where $N$ is some integer).

The Integral Test Statement:

For a series $\sum_{n=1}^{\infty} a_n$, if there is a function $f(x)$ such that:

• $a_n = f(n)$,
• $f(x)$ is continuous, positive, and decreasing for $x \geq N$ (some $N$),
• The improper integral $\int_{N}^{\infty} f(x) \, dx$ converges or diverges,

then:

• If the integral $\int_{N}^{\infty} f(x) \, dx$ converges, the series $\sum_{n=N}^{\infty} a_n$ converges.
• If the integral $\int_{N}^{\infty} f(x) \, dx$ diverges, the series $\sum_{n=N}^{\infty} a_n$ diverges.

Example:

Consider the series: $\sum_{n=1}^{\infty} \frac{1}{n^2}$ We will apply the Integral Test to this series.

1. Define the function: Let $f(x) = \frac{1}{x^2}$. We can see that $f(x)$ is continuous, positive, and decreasing for $x \geq 1$.

2. Compute the integral: We need to evaluate the improper integral: $\int_1^{\infty} \frac{1}{x^2} \, dx$ First, solve the indefinite integral: $\int \frac{1}{x^2} \, dx = -\frac{1}{x}$ Now, evaluate the improper integral: $\int_1^{\infty} \frac{1}{x^2} \, dx = \lim_{b \to \infty} \left[ -\frac{1}{x} \right]_1^b = \lim_{b \to \infty} \left( -\frac{1}{b} + 1 \right)$ As $b \to \infty$, $\frac{1}{b}$ approaches 0, so the integral becomes: $1$ Since the integral converges to 1 (a finite value), the series converges by the Integral Test.

Conclusion:

Using the Integral Test, we have shown that the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges.

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Would you like more examples, or should I explain a different concept related to this?

Here are some related questions:

1. How do you determine whether a function is decreasing for the Integral Test?
2. What happens if the improper integral diverges in the Integral Test?
3. Can the Integral Test be applied to all types of series?
4. How does the comparison test relate to the Integral Test?
5. What other methods can be used to test the convergence of a series?

Tip: When applying the Integral Test, ensure that the function is continuous, positive, and decreasing for large $x$, otherwise the test may not be applicable.