How do I integrate this?

Question

I am supposed to prove this is a divergent series. I'm trying to apply the Integral Test. I think the function is positive, continuous and decreasing on x > or equal to 1, but I can't integrate this. Please help me find the anti-derivative of:

1/(1 + 2ln(x))

When you have an infinite series like this, can you compare logarithms using a direct comparison or limit comparison?

Answer 1

There is no antiderivative in the real number system for 1/(1+2ln(x)). As the other answerer suggested, use the comparison test to show that it diverges.

Answer 2

I would use a comparison test.

Σ[1/(1 + 2ln(x))] > Σ[1/(1 + 2x)] which diverges

Therefore the original series sum diverges.