Series
Infinite series
An infinite series (or just series) of constants ak is defined as a limit of finite series:
∑k=0∞ak=limn→∞∑k=0nak
Geometric Series
A geometric series can be written in the form
∑k=0∞ark=a+ar+ar2+ar3+…
The sum of Geometric series is given by
Sn=1−ra(1−rn+1)
new kind of series from CS370
Doesn’t have to sum to infinity
∑j=0N−1xj=x−1xN−1
where x=1
Telescoping Series
a telescoping series is a series whose general term can be written as the difference of two consecutive terms of a sequence, i.e.
tn=an−an+1
Convergence / Divergence Tests
Conditional vs. Absolute Convergence
A series ∑ak is called absolutely convergent if ∑∣ak∣ converges.
A series ∑ak is said to be conditionally convergent if it converges, but the series ∑∣ak∣ diverges.
To check between for absolute convergence, we use The Ratio Test.