The harmonic series is the infinite sum 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... . Its terms get smaller and approach 0, so it is tempting to think the total should settle near a finite number. Instead, the partial sums keep climbing forever, although very slowly.
This makes the harmonic series a central example in calculus because it shows that terms approaching zero is not enough for convergence.
One way to see the divergence is to group terms and compare each group to a fixed amount, such as 1/2. Another way is to compare the series with the improper integral of 1/x, which also grows without bound. The partial sums grow roughly like ln(n) + gamma, so reaching large totals requires an enormous number of terms.
The harmonic series appears in number theory, physics, probability, computer science, and any setting where many small contributions accumulate.
Key Facts
- The harmonic series is sum from n = 1 to infinity of 1/n = 1 + 1/2 + 1/3 + ... .
- The nth term test says a series can converge only if a_n approaches 0, but a_n -> 0 does not guarantee convergence.
- For the harmonic series, a_n = 1/n and lim as n -> infinity of 1/n = 0.
- The harmonic series diverges: sum from n = 1 to infinity of 1/n = infinity.
- Grouping proof idea: 1/3 + 1/4 > 1/2, 1/5 + 1/6 + 1/7 + 1/8 > 1/2, and each larger group adds more than 1/2.
- Integral test comparison: integral from 1 to infinity of 1/x dx = infinity, so sum from n = 1 to infinity of 1/n diverges.
Vocabulary
- Harmonic series
- The infinite series formed by adding the reciprocals of the positive integers, 1 + 1/2 + 1/3 + 1/4 + ... .
- Partial sum
- The sum of the first n terms of a series, written S_n.
- Divergence
- A series diverges if its partial sums do not approach a finite limit.
- Integral test
- A test that compares an infinite series with an improper integral when the terms come from a positive, decreasing function.
- Comparison test
- A test that determines convergence or divergence by comparing one series to another series whose behavior is already known.
Common Mistakes to Avoid
- Assuming terms going to 0 means the series converges. This is wrong because 1/n approaches 0, but the harmonic series still diverges.
- Confusing individual terms with partial sums. The terms 1/n shrink toward 0, but the partial sums S_n keep increasing without bound.
- Using the nth term test as a convergence test. The nth term test can prove divergence when terms do not approach 0, but it cannot prove convergence when they do.
- Thinking divergence must happen quickly. The harmonic series diverges very slowly, so many early partial sums can look nearly settled even though there is no finite limit.
Practice Questions
- 1 Compute the partial sum S_5 = 1 + 1/2 + 1/3 + 1/4 + 1/5 as a fraction and as a decimal rounded to three places.
- 2 Use grouping to show that the terms from 1/9 through 1/16 add to more than 1/2.
- 3 Explain why the statement 1/n approaches 0, so the harmonic series converges is not a valid argument.