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 not decreasing, while the integral converges and the series diverges (or vice versa), we need to manipulate the behavior of the function for different values of x.

One such example is the function f(x) = 1/x defined on the interval (1, ∞).

1. f(x) is greater than or equal to 0: For any positive value of x, the function 1/x is always positive or zero.

2. f(x) is not decreasing: In this example, f(x) will always be decreasing as x increases because the denominator, x, is getting larger. So, this condition is not satisfied.

Now, let's consider the integral and series:

1. Integral convergence: To check if the integral of f(x) converges, we evaluate the improper integral from 1 to infinity:

∫(1 to ∞) (1/x) dx = ln(x) |(1 to ∞) = ln(∞) - ln(1) = ∞ - 0 = ∞.

Since the integral diverges, this condition is not satisfied.

2. Series divergence: To check if the series ∑ f(x) diverges, we evaluate the sum from 1 to infinity:

∑(1 to ∞) (1/x) = (1/1) + (1/2) + (1/3) + ...,

This series is known as the harmonic series and is a well-known example of a divergent series. Therefore, this condition is satisfied.

In summary, the example of the function f(x) = 1/x on the interval (1, ∞) meets the criteria of being greater than or equal to 0, not decreasing, and having a convergent integral but a divergent series.