Math & Physics Problems Wikia
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Problem

The harmonic series is an infinite series given by the sum of reciprocals.

Part 1: Use the comparison test to prove that the harmonic series is divergent.

Part 2: Use the integral test to prove that the harmonic series is divergent.

Harmonic.jpg

Solution

Part 1 Group the terms in the Harmonic series as follows:

It can be shown that each bracket grouped above is always greater than :

Therefore,

Since

it follows that

.

Part 2


Therefore,

.

Historical Note

Oresme.jpg

This result in Part 1 (using the comparison test) was a proof by the Medieval French theologian, philosopher, and mathematician Nicholas Oresme (c. 1325 - 1382 AD).

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