1.Algebra Booster

Binomial Theorem 6.5
Important Expansions to Remember
(i)
-1 2 3
(1 x) 1 x x x ( 1) r x
r
+ = - + - + + - +
-2 2 3
r r
(ii) (1 + x) = 1 - 2x+ 3x - 4 x + + (- 1) ( r + 1) x +
(iii) (1 + x) –3 = 1 – 3x + 6x 2 – 10x 3 + …
r ( r + 1)( r + 2) r
+ (- 1)
x +
2
(iv) (1 – x) –1 = 1 + x + x 2 + x 3 + … + x r + …
(v)
-2 2 3
(1 - x) = 1 + 2x+ 3x + 4 x + + ( r + 1) x
r +
(vi) (1 – x) –3 = 1 + 3x + 6x 2 + 10x 3 + … +
( r + 1)( r + 2) r
+ x +
2
3. If n ΠN,
(1 – x) –n n n+ 1 2 n+
2 3
= 1 + C1◊ x + C2◊ x + C3◊ x +
n+ r-1
r
+ Cr
◊ x +
Thus, the co-efficient of x r in the expansion of (1 – x) –n
is n+r–1 C r
.
4. If n ΠQ,
nn ( + 1)
(1 – x) –n 2 nn ( + 1)( n+
2) 3
= 1 + nx + x + x +
2! 3!
3.4
5. (1 – x) 3 2 3.4.5 3 3.4.5.6 4
= 1+ 3x+ x + x + x +
1.2 1.2.3 1.2.3.4
- 1
1 1.3
6.
2 2 1.3.5 3 1.3.5.7 4
(1 - x) = 1 + x + x + x + x +
2 2.4 2.4.6 2.4.6.8
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
(xiii)
1
e+
e - 1 1 1
= 1 + + + +
2 2! 4! 6!
1
e-
e - 1 1 1
= 1 + + + +
2 3! 5! 7!
1 1 1
e = Â = Â = Â
n! ( n -1)! ( n - 2)!
Â
n= 0 n= 0 n=
0
1 1
= Â = Â
( n -3)! ( n - k)!
n=
1
Â
n=
2
Â
n= 0 n=
0
1 1 1 1
1 to e 1
n! 2! 3! 4!
1 1 1 1
n! 2! 3! 4!
1 1 1 1 1
( 1)! 1! 2! 3! 4!
n= 0
n +
Â
1 1 1 1
( 2)! 2! 3! 4!
n= 0
n +
Â
n=
0
Â
n=
0
to e 2
to e 1
to e 2
1 1 1 1
e+
e
to
2 n! 2! 4! 6! 2
-1
1 1 1 1
e-
e
to
(2n
- 1)! 1! 3! 5! 2
-1
7. EXPONENTIAL SERIES
Leonhard Euler, the great Swiss Mathematician introduced
and named the number e in his calculus text in 1748 AD.
Definition
1 1 1 1
The sum of the infinite series 1 to
1! 2! 3! 4!
denoted by the number e is
n
Ê 1ˆ
1. e = lim Á1
+
Ë
˜
n n¯
2. e lies between 2 and 3.
2 3
x x x x
3. e 1 to
1! 2! 3!
4. Let a > 0, for all values of x,
x
2 3
x
2 x
e e e
a = 1 + x(log a) + (log a) + (log a)
+
2! 3!
Some Important Expansion to Remember
2 3 4
(i) e
x 1 x
x x x to
2! 3! 4!
(ii)
(iii)
(iv)
2 3 4
-
e
x 1 x
x x x to
2! 3! 4!
x
- x
2 4 6
e + e x x x
1 to
2 2! 4! 6!
x
- x
3 5 7
e - e x x x
x
2 3! 5! 7!
to
8. LOGARITHMIC SERIES
We know that, if ax = n ¤ log a
x = n
Here, a is known as the base of the logarithms.
There are two types of logarithms.
(i) Naperian or Natural Logarithms, where the base is e.
(ii) Common Logarithms, where the base is 10.
Now, we shall obtain an expansion for log e
(1 + x) as a series
in powers of x which is valid only when |x| < 1.
If |x| < 1, then
2 3 4
x x x
log e
(1 + x) = x - + - +
2 3 4
Some Important Expansion to Remember
2 3 4
(i)
x x x
log e(1 + x)
= x- + - +
2 3 4
(ii)
(iii)
(iv)
(v)
2 3 4
x x x
log e(1 - x)
=-x- - - +
2 3 4
3 5 7
Ê1
+ xˆ Ê x x x ˆ
logÁ = 2Áx
+ + + + ˜
Ë1-
x˜ ¯ Ë 3 5 7 ¯
1 1 1 1 1
log 2 = 1 - + - + - +
2 3 4 5 6
2 4 6
2
Ê x x x
log(1 - x ) =- 2Á
+ + +
Ë 2 4 6
ˆ
˜
¯