how do i show that the the sum as n approaces infitiy of the series defined by an = ln(1+1/n) is divergent?

Question

calculus series divergence mathematics

Answer 1

a[n] = ln(1 + (1/n))

You can use the comparison test.

ln(1 + (1/n)) > ln(1/n)

If we calculate
lim (ln(1/n))
n -> infinity

This is the same as calculating

lim (-ln(n))
n -> infinity

As n goes to infinity, -ln(n) approaches negative infinity, which means the sequence defined by a[n] = ln(1/n) diverges.

Since ln(1/n) diverges, it follows that ln(1 + (1/n)) diverges too.

Answer 2

If you take b_n=1/n lim ln(1+1/n) /(1/n) =1 so a_n and b_n are of the same class but b_n=1/n is divergent(Harmonic series) so a_n is divergent

Answer 3

Use the fact that the the sum of logs is the log of the product

Answer 4

Try the integral test.