, where is the term of a sequence, is called a Series.
If the number of terms is finite, then it is called Finite Series and if the number of terms is infinite, it is called Infinite Series.
Finite series -:
For a given , is a Finite Series.
Infinite Series -:
An expression the which involves addition of infinitely many real numbers, which we clarify to find the the sum of infinitely many real numbers.
Partial Sums of an Infinite Series -:
For an infinite series , a sum is called the partial sum of the series;
, and so on.
Consequently, given a sequence , we obtain another related sequence , which is the sequence of partial sums of the infinite series .
Sum of an Infinite Series -:
Let be the sequence of partial sums of an infinite series .
is convergent, if there exists , such that .
If is convergent, then is said to be convergent and is called its sum.
is convergent, if the sequence of partial sums is convergent.
If does not converge, i.e. does not exist, then the series is called a Divergent Series, and we say diverges.