Definition
An harmonic series is a series in the form:
Divergence
A harmonic series is divergent. This can be demonstrated through the Integral test or other method
Example
Show that the following series is divergent:
Solution
As written above the harmonic series can be written as:
Let's replace the sum of all the terms after the third term by:
Then we have:
Let's solve this equation for
The series in the second side is divergent as an harmonic series. Being an harmonic series, its value can be infinite or it doesn't have any value at all. Substracting a finite value from infinity or an indefinite value doesn't change infinity or the indefinite value. The harmonic series in the second side remains divergent. Therefore the series in the first side is divergent,
Practice
Show that the following series is divergent:
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