Give an Example where f(x) is greater than or equal to 0 but f(x) is not decreasing and the integral converges while the series diverges or vice versa.
To find an example where f(x) is greater than or equal to 0 but f(x) is not decreasing, while the integral converges and the series diverges (or vice versa), we can consider the following function:
Let f(x) = 1/x for x > 1 and 0 for 0 ≤ x ≤ 1.
1. f(x) is greater than or equal to 0 for all x because 1/x is positive for x > 1 and 0 for 0 ≤ x ≤ 1.
2. f(x) is not decreasing because it increases as x approaches 0.
3. The integral of f(x) from 1 to infinity converges:
∫(1 to ∞) 1/x dx = ln|x| (from 1 to ∞)
= ln(∞) - ln(1)
= ∞ - 0
= ∞
Since the integral diverges, it means the integral of f(x) converges.
4. However, the series defined by the function f(x) diverges:
∑(n=1 to ∞) 1/n diverges.
This is a well-known result in calculus, known as the harmonic series, which diverges.
Thus, we have found an example where f(x) is greater than or equal to 0, f(x) is not decreasing, the integral converges, but the series diverges.