The integral test

Question

Use the Integral Test to determine whether the series is convergent or divergent.

∑(from n=1 to ∞) 8/(7n-1)

Evaluate the following integral.

∫ (from 1 to ∞) 8/(7x-1) dx

Answer 1

The integral test says the integral and the series both converge or both diverge. This is a great series to use the integral test because this is a function that you can find an anti-derivative for (which is rare). Try this integral with u-sub.

∫₁^∞ 8/(7x-1) dx

u = 7x-1

du = 7 dx

∫₆^∞ 8/(7u) du

This is an improper integral. It depends how strict your professor is but you might need to write the integral with a limit outside. As you get into other courses that use calculus they will drop this formality.

8/7 ln(u) |₆^∞ = ∞

So the series diverges.

Answer 2

Here you go! Video should download shortly. I hope this helps!