Special Series
Geometric Series¶
A geometric series is the summation of increasingly diminishing terms. $ a $ is the first term of the sequence, and $ r $ is the common ratio of the series.
\[
S = \sum_{n=1}^{\infty} ar^{n-1} = a + ar + ar^2 + ar^3 + ...
\]
The sum of a convergent geometric series is:
\[
S = a_1 \cdot \frac{1 - q^n}{1 - q}
\]
Harmonic Series¶
The harmonic series has the form of: $$ \sum_{n=1}^{\infty} \frac{1}{n} $$ A harmonic series does not converge.
Hyperharmonic (p-series)¶
The hyperharmonic or p-series has the form of $$ \sum_{n=1}^{\infty} \frac{1}{n^p} $$ If $ p > 1 $ then the series will converge. We say that a series converges by the p-test.