SHORT NOTES MODULE 3 : SERIES
POWER SERIES: Power series centred at c or Power series in x – c
Infinite series → infinite terms
c → centre of power series
Index below – 1st index above – last index a0, a1, a2, …, an → coefficients
TAYLOR SERIES AND MACLAURIN SERIES FORMULA:
n=0
n f ( ) (c )
n!
(x − c)
n
1 2 f ( ) (c ) f ( ) (c ) 0) ( = f (c ) + (x − c) + ( x − c )2 + 1! 2!
TAYLOR SERIES
c=0
n =0
n f ( ) (0)
n!
1 2 f ( ) (0) f ( ) (0) 2 0) ( x = f (0) + x+ x + 1! 2! n
MACLAURIN SERIES
CONSTRUCT A TAYLOR/MACLAURIN SERIES TAYLOR SERIES (MACLAURIN SERIES WHEN c = 0) Series of function, f(x)
substitute the values of derivative and c into the formula and simplify
CONSTRUCTING TAYLOR/ MACLAURIN SERIES
substitute c into every derivative
identify the centre, c and the required index, n (or number of term)
differentiate up until the nth order
COMMON MACLAURIN SERIES
e = x
n =0
xn x 2 x3 = 1+ x + + + n! 2! 3!
ln (1 + x ) =
n =0
( −1)n+1 xn n
=x−
x 2 x3 x 4 + − + 2 3 4
x
1 = 1− x
n
= 1 + x + x 2 + x3 +
n =0
sin x =
( −1)n x2n+1 = x − x3 + x5 − x7 + 3! 5! 7! ( 2n + 1)! n =0
n −1) x2n ( cos x = ( 2n) ! n =0
= 1−
x2 x 4 x6 + − + 2! 4! 6!
BINOMIAL SERIES
BINOMIAL COEFFICIENT
1 1 PASCAL TRIANGLE
1
1 2
1
3
1 3
1
⋮
BINOMIAL COEFFICIENT FOR (a+b)n ; n∈ ℝ COMBINATORIAL FORMULA
nC r
=
𝑛 = 𝑟! 𝑟
𝑛! 𝑛 −𝑟 !
BINOMIAL THEOREM
BINOMIAL EXPANSION USING BINOMIAL THEOREM
(1 + x )n
; n
, − 1 x 1
Finite series if n∈ ℕ Valid interval of x: -1 < x < 1
(a + b)n
; n
, a,b
Finite series Number of terms = n + 1
(a + b)
n
n
=
n =0
n r n−r a b r
n n n = a0bn + abn−1 + a2bn−2 + 0 1 2
( a + b )n =
n
n =0
(1 + x )n = 1 + nx +
n n−r r a b r
n + anb0 n
OR
n n n = anb0 + an−1b1 + an−2b2 + 0 1 2
Specific term of binomial expansion, kth term: n Tk = Tr +1 = ar bn−r r
OR n Tk = Tr +1 = an−r br r
n + a0bn n
n (n − 1) 2!
x2 +
n (n − 1)(n − 2) 3!
x3 +