2.4 The Harmonic Series (continued)
In the fourteenth century, Nicole Oresme had pointed out that this series grows without limit. He showed that we can split it into sums of fractions adding up to at least 1/2, with as many of these sums as we want:
How fast does the harmonic series grow? Equation (2.4.5) shows that if we add the first terms of the harmonic series, we get a sum that is approximately 1 + n/2. This suggests that the size of the sum of the first n terms of the harmonic series should be related to a logarithm.
It was Leonhard Euler who, in 1734, first established the exact connection between the harmonic series and the natural logarithm. Following his path, we compare the natural logarithm of n to the sum of the first n–1 terms of the harmonic series. We denote the difference by
The sequence appears to be increasing, and it seems to approach a value a little above 1/2. In fact, it does approach a constant value known as Euler's constant and denoted by the Greek letter .
We can interpret geometrically. Recall that ln n is the area under the curve y = 1/x from x = 1 to n. The sum is the area under from x = 1 to n (see figure 2.2), where denotes the greatest integer less than or equal to x, read as the floor of x.)
Figure 2.2 The area between the graphs of ln x and 1/.
The quantity is the area between these graphs for This confirms that the sequence is, in fact, increasing. We can approximate by approximating the area as a sequence of triangles.
The sum of the areas of the triangles is
This sum approaches 1/2 as n gets larger. The area between the graphs is larger than 1/2 because the graph of y = 1/x is concave up. But this gives us some idea of the probable size of .
Macalester Home | Directory | Site Map | Search
About Macalester | Academic Programs | Admissions | Alumni & Parents | Athletics
Administrative Offices | Information Services | News & Events | Student Services
Macalester College · 1600 Grand Avenue, St. Paul, MN 55105 · 651-696-6000