Power 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.

Def; If the sequence {S_n } of partial sums converges to some point s , written ?lim??(n??)??S_n ?=s, then the series ?_(k=1)^??a_k =a_1+a_2+a_3+? is said to converges and have the sum s. In this case we write
?_(k=1)^??a_k =s
Example: Determine whether the series
1+1/3+1/9+1/27+1/81+?+(1/3)^n+?=?_(k=0)^??(1/3)^k
converges or diverge, then find its sum.
We form the sequence of partial sums:
S_0=1
S_1=1+1/3
S_2=1+1/3+1/9
S_3=1+1/3+1/9+1/27
and so forth. In general
S_n=1+1/3+1/9+1/27+1/81+?+(1/3)^n
Multiplying S_n by 1/3 gives
?1/3 S?_n=1/3+1/9+1/27+1/81+?+(1/3)^(n+1)
S_n-?1/3 S?_n=1-(1/3)^(n+1)
?2/3 S?_n=1-(1/3)^(n+1)
S_n=3/2 [1-(1/3)^(n+1) ]
?lim??(n??)??S_n ?=3/2 [1-(1/3)^(n+1) ]=3/2
Thus, the series converges and has the sum 3/2.