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Series formulas are arithmetic’s and geometric series, special power series, Taylor and McLaurin series.
Typology: Cheat Sheet
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Definitions:
First term: a 1 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 = a 1 (^) + (^) ( n − (^1) ) d
1 1 2
i i i
a a a −^ +
1 2
n n
a a S n
(^2 1) ( 1 )
2
n
a n d S n
Geometric Series Formulas:
1 1
n a n a q
− = ⋅
ai = ai (^) − 1 ⋅ ai + 1
1 1
n n
a q a S q
1 (^1 )
1
n
n
a q S q
1
1
fo 1 1
a S q
r − q −
Powers of Natural Numbers
( ) 1
n
k
∑^ =^ +
( )( )
2
1
n
k
∑^ =^ +^ +
( )
3 2 2
1
n
k
∑^ =^ +
Special Power Series
( )
x x x x x
= + + + + for − < < −
( )
x x x x x
= − + − + for − < <
2 3 1... 2! 3!
x x^ x e = + x + + +
( ) ( )
2 3 4 5 ln 1... 1 1 2 3 4 5
x x x x
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 x x x
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
Definition:
( )
( ) ( )
( )
2 1 1 ( )( ) ( ) ( ) ( ) ( )... 2! 1!
n^ n
n
f a x a^ f^ a^ x^ a f x f a f a x a R n
− − ′′ (^) − − = + ′ − + + + + −
( ) ( )( )
( ) ( )( ) ( )
( )
1
n^ n
n
n n
n
f x a R Lagrange s form a x n
f x x a R Cauch s form a x n
−
This result holds if f(x) has continuous derivatives of order n at last. If lim (^) n 0 n
→∞
= , the infinite series obtained is called
Taylor series for f(x) about x = a. If a = 0 the series is often called a Maclaurin series.
Binomial series
( )
1 (^ )^2 2 (^ )(^ ) 3 3
1 2 2 3 3
n (^) n n n n
n n n n
n n n n n a x a na x a x a x
n n n a a x a x a x
− − −
− − −
Special cases:
( )
(^1 2 3 ) 1 x 1 x x x x ... 1 x 1
−
( )
(^2 2 3 ) 1 x 1 2 x 3 x 4 x 5 x ... 1 x 1
−
( )
(^3 2 3 ) 1 x 1 3 x 6 x 10 x 15 x. .. 1 x 1
−
( )
1 2 1 1 3^2 1 3 5^3 1 1 ... 2 2 4 2 4 6
x x x x 1 x 1
( )
1 2 1 1 2 1 3^3 1 1 ... 2 2 4 2 4 6
x x x x 1 x 1
Series for exponential and logarithmic functions
2 3 1 ... 2! 3!
x x^ x e = + x + + +
( ) ( )
2 3 ln ln 1 ln ... 2! 3!
x x^ a^ x^ a a = + x a + + +
( )
2 3 4 ln 1 ... 2 3 4
x x x
( )
2 3 1 1 1 1 1 ln 1 ... 2 3
x x x x x x x
x