P-Series and harmonic Series

P-Series:

A p-series is a series of the form:


Where p is a real number .
The p-series converges if p>1
The p-series diverges if

Example:
Test the convergence or divergence of the series

Solution:
We have:


The p-series converges .

Example:
. Test the convergence or divergence of the series


Solution:
We have:



The p-series diverges .

Harmonic Series:
A series of the form:

is called a harmonic series.
The harmonic series diverge.

Some Examples about Geometric Series

Example:
Test the convergence or divergence of the series
Solution:
We have:



The series is convergent.

Example:
The geometric series with a = 1/9 and r = 1/3 :


Example:
The geometric series with a = 4 and r = -1/2 :



Repeating Decimals:
We use geometric series to explain why repeating decimals represent rational numbers.
Example:
Express the repeating decimal 5.23 23 23….. as the ratio of two integers.
Solution:
5.23 23 23…..