series

Series المتسلسلات

Def; An Infinite Series is an expression of the form
a_1+a_2+a_3+?+a_k+?
Or we use the summation notation
?_(k=1)^??a_k

It is important to understand the difference between a sequence and a series. As explained in chapter 1, a sequence is simply a set of numbers arranged in a certain order. On the other hand, a series is a sum.
The sequence

{a_k }_(k=1)^?={a_1,a_2,a_3,…,a_k,…}
The series
?_(k=1)^??a_k =a_1+a_2+a_3+?+a_k+?
The quantities a_1,a_2,a_3,…are called the terms of the series. a_k is referred to as the general term.


Examples:
?_(k=0)^??1/(k^2+4)=1/4+1/5+1/8+1/13+1/20+?

?_(k=4)^??(k+1)/k^2 =5/16+6/25+7/36+8/49+9/64+?

?_(k=2)^??1/(k lnk)=1/2ln2+1/3ln3+1/4ln4+1/5ln5+1/6ln6+?

Convergence of Series:
If we let S_n be the sum of the first n terms of the series
?_(k=1)^??a_k =a_1+a_2+a_3+?+a_k+?,
then
S_1=a_1
S_2=a_1+a_2
S_3=a_1+a_2+a_3
and so forth. In general
S_n=a_1+a_2+a_3+?+a_n
The numbers S_1,S_2,S_3,…,S_n,… form a sequence {S_n } called the sequence of partial sums of the series above.