Why does the Harmonic Series diverge?

Answer 1

The Harmonic Series diverges because the sum of its terms grows without bound as more terms are added. Mathematically, it can be proven using the integral test or the comparison test that the sum of the reciprocals of natural numbers (1/n) diverges. This means that the series does not converge to a finite value but instead increases indefinitely as more terms are added.

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Answer 2

Charles Anctil

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.

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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.