Some Examples about Geometric Series,P-Series and Harmonic Series
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…..
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.
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