Arithmetic series are special mathematical series . An arithmetic series is the sequence whose terms are the sum of the first terms (the partial sums ) of an arithmetic sequence . Arithmetic series are generally divergent. The partial sums, which are also called finite arithmetic series , are of particular interest .
In an arithmetic sequence , the -th term can be used as
write, where is the (constant) difference between two consecutive terms.
The -th partial sum of an arithmetic series results in
General molecular formula
There is a simple formula for calculating the partial sums (or the finite arithmetic series):
In the last form, the formula is particularly easy to remember: The sum of a finite arithmetic sequence is the number of terms multiplied by the arithmetic mean of the first and the last term.
The proof of this equation is often used as a first example of the application of the complete induction method.
The Gaussian empirical formula applies to the sum of the first natural numbers
and for the sum of the first odd natural numbers
with , .
- Eric W. Weisstein : Arithmetic Series . In: MathWorld (English).