If (constant), for , then is called an Arithmetic Sequence or an Arithmetic Progression and the series is called an Arithmetic Series. The constant is known as the common difference which may be positive or negative.
Partial Sum of an Arithmetic Series -:
If is an Arithmetic series, then which is same as , where ,
Common difference , then of the finite series .
Therefore or according as or .
If or according as or .
This shows that an Arithmetic Series always diverges, except the special case when .
In this case for all so that sum of the corresponding Arithmetic Series is zero.
Sum of the First all Counting Number -: