does the series 1/n from n=0 to infinity converge or diverge? what convergence test is this?
To determine whether the series 1/n from n=0 to infinity converges or diverges, we can apply the convergence test called the Harmonic Series Test.
The Harmonic Series Test states that if the terms of a series do not approach zero, then the series diverges. In this case, the terms of the series 1/n do approach zero as n approaches infinity, so the Harmonic Series Test does not provide a conclusive answer.
Instead, we can use the Integral Test, another convergence test usually used for series with terms that are positive, continuous, and decreasing. For this test, let's consider the function f(x) = 1/x and integrate it from 1 to infinity:
∫(1/x) dx from 1 to infinity
The integral of 1/x is ln(x), and applying the limits of integration, we get:
∫(1/x) dx = ln(x) | from 1 to ∞
= ln(∞) - ln(1)
= ∞ - 0
= ∞
Since the integral of 1/n from 1 to infinity diverges, this implies that the series 1/n also diverges.
In conclusion, the series 1/n from n=0 to infinity diverges, and we can determine this by applying the Integral Test.