How do you Find the sum of the harmonic series?

Answer 1

The harmonic series diverges. #sum_{n=1}^{infty}1/n=infty#

Let us show this by the comparison test. #sum_{n=1}^{infty}1/n=1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+cdots# by grouping terms, #=1+1/2+(1/3+1/4)+(1/5+1/6+1/7+1/8)+cdots# by replacing the terms in each group by the smallest term in the group, #>1+1/2+(1/4+1/4)+(1/8+1/8+1/8+1/8)+cdots# #=1+1/2+1/2+1/2+cdots# since there are infinitly many #1/2#'s, #=infty#

Since the above shows that the harmonic series is larger that the divergent series, we may conclude that the harmonic series is also divergent by the comparison test.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.