Mathematics REVISION OF Exponential and Logarithmic Series FOR NDA

Exponential Series

The sum of the series `1 + 1/(1!) + 1/(2!) + 1/(3!) +........` is denoted by the number `e` which is an irrational number lying between `2` and `3`.

i.e., `lim_(n->oo) (1+1/n)^n =e = lim_(x->0) (1+x)^(1/x)`

1. `e^x = 1+x + (x^2)/(2!) + (x^3)/(3!) + .......` for all real values of `x, e^x = sum_(n=0)^(oo) (x^n)/(n!)`

2. For `a > 0 , a^x =1 + x/(1!) (log_e a) + (x^2)/(2!) (log_e a)^2 + (x^3 )/(3! ) (log_e a)^3 + ........= e^(x log_e a )`

3. `e^(-x) = 1- x/(1!) + (x^2)/(2!) - (x^3)/(3!) + ........... = sum_(n=0)^(oo) (-1)^n (x^n)/(n!)`

4. `(e^x + e^(-x) )/2 =1 + (x^2)/(2!) + (x^4)/( 4!) + (x^6)/(6!) + ..... = sum_(n=0)^(oo) (x^(2n))/( 2n!)`

5. `( e^x - e^(-x) )/2 = x/(1!) + (x^3)/(3!) + (x^5)/(5!) + .... = sum_(n=0)^(oo) (x^(2n +1 ) )/( (2n +1)! )`

6. `( e+ e^(-1) )/2 = 1/(1!) +1/(2!) +1/(4!) + 1/(6!) + ...... = sum_(n=0)^(oo) 1/(2n!)`

7. `(e-e^(-1) )/2 = 1/(1!) +1/(3!) + 1/(5!) + ....= sum_(n=0 )^oo 1/( (2n+1)! )`

8. `sum_(n=0)^(oo) 1/(n!) = sum_(n=1)^oo 1/( (n-1)! ) = sum_(n=2)^oo 1/( (n-2)! ) = e`

9. `e= 1 + 1/(1!) + 1/(2!) + 1/(3!) + 1/(4!) + ........`

10. `e^(-1) = 1- 1/(1!) + 1/(2!) - 1/(3!) +1/(4!) - ......`

11. `lim_(n->oo) (1+1/n)^n = lim_(n->oo) (1+n)^(1/n) = e`

Logarithm

If `a` is a positive real number other than 1 and `a^x = m `, then `x` is called the `"logarithm of m to the base a,"` written as `log_a m`.
In `log_a m, m` should always be positive.

(i) If `m < 0`, then `log_a m` will be imaginary and if `m = 0`,
then `log_a m` will be meaningless.

(ii) `log_a m` exists, if `m , a > 0` and `a ne 1`

Properties of Logarithm

1. `a^(log_(a) x) = x ; a ne 0 , ne 1 , x > 0`

2. `a^(log_(b) x ) = x^(log_b a) ; a , b > 0 , ne 1 , x > 0`

3. `log_a a =1, a > 0 , ne 1`

4. `log_a x = 1/(log_x a) ; x , a > 0 , ne 1`

5. `log_a x = (log_b x)/( log_b a) ; a , b > 0 , ne 1 , x > 0`

6. For `m, n > 0` and `a > 0,` `ne 1`, then

(i) `log_a (m * n) = log_a m + log_a n`

(ii) `log_a (m/n) = log_a m - log_a n`

(iii) `log_a (m^n) = n log_a m`

7. for `x > 0 , a > 0 , ne 1`

(i) `log_(a^n) (x) = 1/n log_a x`

(ii) `log_(a^n) x^m = (m/n) log_a x`

8. for `x > y > 0`

(i) `log_a x > log_a y`, if `a > 1`
(ii) `log_a x < log_a y ` if ` 0 < a < 1`

9. If `a > 1` and `x > 0`, then

(i) `log_a x > p => x > a^p`

(ii) `0 < log_a x < p => 0 < x < a^p`

10. If ` 0 < a < 1`, then

(i) `log_a x > p => 0 < x < a^p`

(ii) `0 < log_a x < p => a^p < x < 1`

Logarithmic Series

1. `log_e (1+x) = x- (x^2)/2 + (x^3)/3 - (x^4)/4 + ....`

2. `log_e (1-x) = - (x+ x^2/2 + x^3/3 + x^4/4 + .....)`, where `-1 le x`

3. `log_e ( (1+x)/(1-x) ) = 2 (x+ x^3/3 + x^5/5 + .......)`

4. `log [ (1+x) (1-x) ] =2 - ( x^2/2 + x^4/4 + x^6/6 + ...)`

5. `log_e (n+1) - log_e (n-1) = 2 (1/n + 1/(3n^3) + 1/(5n^5) + ..... )`

6. `log_e 2 = (1-1/2 +1/3 -1/4 + ......) = 0.693147`

or `0.61 < log 2 < 0.76`

`log_e 2 = 1/(1*2) + 1/(3*4) + 1/(5*6) + ............`