Question

State the Comparison Theorem for improper integrals.

Short answer

for then \(f(x) \ge g(x) \ge 0\)

If \(f(x)\)is convergent then \(g(x)\) is also Convergent and if \(g(x)\) is divergent then \(f(x)\)is also divergent

Step-by-step solution

Step-1: Explanation

If we have an Improper integral then we use Comparison Theorem

In this theorem we take a similar function \(g\) of a given function \(f\) , then we check and calculate values for the function \(f\) . If \(f\) is Convergent then we assume that \(g\) is also convergent and vice-versa. Similarly, if \(f\) Divergent then \(g\) is also divergent and vice-versa. (Here function \(f\) and \(g\) are continuous function in the given domain)

Step-2:  Comparison Theorem for Improper Integrals

Suppose that two functions \(f(x)\) and \(g(x)\) are continuous in a given domain

\(f(x) \ge g(x) \ge 0\)for \(x \ge a\) then

\((i)\) If\(\int_a^\infty {f(x)} dx\)is convergent, then \(\int_a^\infty {g(x)} dx\) is convergent.

\((ii)\) If\(\int_a^\infty {g(x)} dx\)is divergent, then\(\int_a^\infty {f(x)} dx\)is divergent.