Series Formulas
Series Formulas
1. Arithmetic and Geometric Series 2. Special Power Series
Definitions:
First term: a1 Nth term: an Number of terms in the series: n Sum of the first n terms: Sn Difference between successive terms: d Common ratio: q Sum to infinity: S
Arithmetic Series Formulas:
an = a1 + (n -1) d
ai
=
ai -1
+ ai+1 2
Sn
=
a1
+ an 2
n
Sn
=
2a1
+ (n -1) d
2
n
Geometric Series Formulas:
an = a1 qn-1
ai = ai-1 ai+1
Sn
=
an q - a1 q -1
( ) Sn
=
a1
qn -1 q -1
S = a1 1- q
for -1 < q < 1
Powers of Natural Numbers
n k = 1 n (n +1)
k =1
2
n k 2 = 1 n (n +1)(2n +1)
k =1
6
n k 3 = 1 n2 ( n +1)2
k =1
4
Special Power Series
1 = 1+ x + x2 + x3 + . . . ( for : -1 < x < 1)
1- x
1 = 1- x + x2 - x3 + . . . ( for : -1 < x < 1)
1+ x
ex =1+ x + x2 + x3 + . . . 2! 3!
ln(1+ x) = x - x2 + x3 - x4 + x5 . . . ( for : -1< x ................
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