Harmonic — Definition, Formula & Examples

Harmonic refers to a sequence or series formed by taking the reciprocals of the positive integers or, more generally, the reciprocals of an arithmetic sequence. The most famous example is the harmonic series:

1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots

A harmonic sequence is a sequence whose terms are the reciprocals of an arithmetic sequence. If $a_1, a_2, a_3, \ldots$ is an arithmetic sequence with nonzero terms, then $\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \ldots$ is the corresponding harmonic sequence. The harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ is the sum of the standard harmonic sequence and is a classic example of a divergent series.

Key Formula

a_n = \frac{1}{a + (n-1)d}

Where:

$a_n$ = The $n$th term of the harmonic sequence
$a$ = The first term of the underlying arithmetic sequence
$d$ = The common difference of the underlying arithmetic sequence
$n$ = The term number (positive integer)

How It Works

To build a harmonic sequence, start with any arithmetic sequence (like

2, 4, 6, 8, \ldots

) and take the reciprocal of each term to get

\frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, \ldots

The terms of a harmonic sequence always decrease toward zero, but this does not guarantee that summing them produces a finite result. The harmonic series

\sum \frac{1}{n}

grows without bound — it diverges — even though the individual terms shrink to zero. This makes it a crucial counterexample in the study of convergence.

Worked Example

Problem: Write the first 5 terms of the harmonic sequence whose underlying arithmetic sequence is

3, 5, 7, 9, 11, \ldots

and find the partial sum of those 5 terms.

Step 1: Identify the arithmetic sequence and take reciprocals.

\frac{1}{3},\; \frac{1}{5},\; \frac{1}{7},\; \frac{1}{9},\; \frac{1}{11}

Step 2: Add the five terms using a common denominator or decimals.

S_5 = \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{9} + \frac{1}{11} \approx 0.3333 + 0.2000 + 0.1429 + 0.1111 + 0.0909

Step 3: Compute the sum.

S_5 \approx 0.8782

Answer: The first 5 terms are

\frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \frac{1}{9}, \frac{1}{11}

, and their sum is approximately

0.878

Why It Matters

The harmonic series appears in AP Calculus and college analysis courses as the standard example of a series whose terms approach zero yet still diverges. Understanding it is essential for applying convergence tests like the p-series test and the comparison test.

Common Mistakes

Mistake: Assuming the harmonic series converges because its terms approach zero.

Correction: Terms approaching zero is necessary but not sufficient for convergence. The harmonic series

\sum 1/n

diverges, which you can prove using the comparison test or the integral test.