short notes module 3 series

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 +