Series
Series is a sum of the terms of a sequence denoted by
![](https://media1.shmoop.com/images/calculus/calc_series_glossary_latek_1.png)
.
Alternating Series
An alternating series is one whose terms alternate between a positive and a negative sign.
Partial Sum
The sum of
n terms of a series is called the
nth partial sum denoted by S
n.
Convergent Series
When the sequence of partial sums S
n converges the series is convergent.
Convergent Sequence
If the limit of a sequence exists, the sequence is convergent.
Arithmetic Series
This occurs when the terms of a series form an arithmetic sequence.
Geometric Series
This occurs when the terms of a series form a geometric sequence.
Divergence Test
When the
nth term of a series does not approach 0, the series is divergent.
Alternating Series Test
For an alternating series
![](https://media1.shmoop.com/images/calculus/calc_series_glossary_latek_2.png)
, if the series
![](https://media1.shmoop.com/images/calculus/calc_series_glossary_latek_3.png)
is decreasing and passes the divergent test, the alternating series converges.
Ratio Test
For the series
![](https://media1.shmoop.com/images/calculus/calc_series_glossary_latek_4.png )
, if the limit
![](https://media1.shmoop.com/images/calculus/calc_series_glossary_latek_5.png )
is less than 1, the series converges. If
L > 1, the series diverges and for
L = 1, the test is inconclusive.