Series Formulas
1. Arithmetic and Geometric 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:
(
)
1
1
n
a a n d
=+−
1 1
2
i i
i
a a
a
− +
+
=
1
2
n
n
a a
S n
+
= ⋅
(
)
1
2 1
2
n
a n d
S n
+ −
= ⋅
Geometric Series Formulas:
1
1
n
n
a a q
−
= ⋅
1 1
i i i
a a a
− +
= ⋅
1
1
n
n
a q a
Sq
−
=−
(
)
1
1
1
n
n
a q
Sq
−
=−
1
1
1 1
fo
a
Sqr q−
−
< <
=
2. Special Power Series
Powers of Natural Numbers
( )
1
1
1
2
n
k
k n n
=
= +
∑
( )( )
2
1
1
1 2 1
6
n
k
k n n n
=
= + +
∑
( )
2
3 2
1
1
1
4
n
k
k n n
=
= +
∑
Special Power Series
( )
2 3
1
1 . . . 1 1
:
1
x x x x
x
for
= + + + + − < <
−
( )
2 3
1
1 . . . 1 1
:
1
x x x x
x
for
= − + − + − < <
+
2 3
1 . . .
2! 3!
xx x
e x= + + + +
( ) ( )
2 3 4 5
ln 1 . . . 1 1
2 3 4 5
:
x x x x
xorx xf
+ = − + − + − < <
3 5 7 9
sin . . .
3! 5! 7! 9!
x x x x
x x= − + − +
2 4 6 8
cos 1 . . .
2! 4! 6! 8!
x x x x
x= − + − +
3 5 7
2 17
tan . . .
3 15 315 2 2
:for
x x x
xx x
π π
= + + + + − < <
3 5 7 9
sinh . . .
3! 5! 7! 9!
x x x x
x x= + + + +
2 4 6 8
cosh 1 . . .
2! 4! 6! 8!
x x x x
x= + + + +
3 5 7
2 17
tan . . .
3 15 315 2 2
:for
x x x
x x x
π π
= − + − + − < <
