Summary Series formulas

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Series Formulas
1. Arithmetic and Geometric Series 2. Special Power Series
Definitions: Powers of Natural Numbers
First term: a1 n
1
Nth term: an ∑ k = 2 n ( n + 1)
k =1
Number of terms in the series: n
n
1
Sum of the first n terms: Sn
Difference between successive terms: d
∑k
k =1
2
=
6
n ( n + 1)( 2n + 1)

Common ratio: q n
1 2
∑k
2
Sum to infinity: S
3
= n ( n + 1)
k =1 4
Arithmetic Series Formulas:
Special Power Series
an = a1 + ( n − 1) d
1
a + ai +1
= 1 + x + x 2 + x3 + . . . ( for : − 1 < x < 1)
ai = i −1 1− x
2 1
a + an
= 1 − x + x2 − x3 + . . . ( for : − 1 < x < 1)
Sn = 1 ⋅n 1+ x
2 x 2 x3
ex = 1 + x + + + ...
2a1 + ( n − 1) d 2! 3!
Sn = ⋅n
2 x2 x3 x 4 x5
ln (1+ x ) = x − + − + ... ( for : −1 < x < 1)
Geometric Series Formulas: 2 3 4 5
an = a1 ⋅ q n−1 x3 x5 x 7 x9
sin x = x − + − + ...
3! 5! 7! 9!
ai = ai −1 ⋅ ai +1
x 2 x 4 x6 x8
an q − a1 cos x = 1 − + − + ...
Sn = 2! 4! 6! 8!
q −1 x3 2x5 17x7  π π
tan x = x + + + + ...  for : − < x < 
Sn =
( n
a1 q − 1 ) 3 15 315  2 2
q −1 x3 x5 x 7 x 9
sinh x = x + + + + ...
a1 3! 5! 7! 9!
S= for − 1 < q < 1
1− q x 2 x 4 x6 x8
cosh x = 1 + + + + ...
2! 4! 6! 8!
x3 2x5 17x7  π π
tan x = x − + − +...  for : − < x < 
3 15 315  2 2