Exponential And Logarithmic Series

Exponential Series

$e^x$ = 1+$\frac{X}{1!}$+$\frac{X^2}{2!}$+$\frac{X^3}{3!}$+.........$\infty$= $\underset{n=0}{\overset{\infty }{\mathrm{\Sigma }}}$ $\frac{x^n}{n!}$
Holds for all real valves of x.

Particular cases

1. $e^{-x}$= 1-$\frac{x}{1!}$+$\frac{x^2}{2!}$+$\frac{x^3}{3!}$+......$\infty$

2. $\frac{e^x+e^{-x}}{2}$= 1+ $\frac{x^2}{2!}$+ $\frac{x^4}{4!}$+.......$\infty$

3. $\frac{e^x-e^{-x}}{2}$= x + $\frac{x^3}{3!}$+$\frac{x^5}{5!}$+........$\infty$

4. e = 1 + $\frac{1}{1!}$+$\frac{1}{2!}$+.......$\infty$

5. $e^{-1}$= 1- $\frac{1}{1!}$+$\frac{1}{2!}$+$\frac{1}{3!}$+......$\infty$

6. $\frac{e-e^{-1}}{2}$= $\frac{1}{1!}$+$\frac{1}{3!}$+$\frac{1}{5!}$......$\infty$

7. $\frac{e^1+e^{-1}}{2}$=1 + $\frac{1}{2!}$ + $\frac{1}{4!}$+.....$\infty$

8. $a^x$= 1+ x${\log }_{e}a$+ $\frac{{\left(x{\log }_{e}a\right)}^2}{2!}$+.......$\infty$

9. $\underset{n=0}{\overset{\infty }{\mathrm{\Sigma }}}$ $\frac{1}{n!}$= e

10. $\underset{n=0}{\overset{\infty }{\mathrm{\Sigma }}}$${(-1)}^n$ $\frac{1}{n!}$=$\frac{1}{e}$

e= 2.707 (correct upto three decimal places)

Logarithmic Series

${\log }_e$ (1+x)= x - $\frac{x^2}{2}$+ $\frac{x^3}{3}$-$\frac{x^4}{4}$+.......$\infty$, where |x| $<$1

1. ${\log }_e$ (1-x)= x - $\frac{x^2}{2}$- $\frac{x^3}{3}$.......$\infty$

2. -log (1-x)= x + $\frac{x^2}{2}$+ $\frac{x^3}{3}$+.......$\infty$

3. 1/2 log $\frac{1+x}{1-x}$= x + $\frac{x^3}{3}$+ $\frac{x^5}{5}$+....$\infty$

4. 1/2log[(1+x) (1-x)] = $\frac{x^2}{2}$+$\frac{x^4}{4}$+......$\infty$

5. log2 = 1- $\frac{1}{2}$+$\frac{1}{3}$- $\frac{1}{4}$+.......$\infty$

6. $e^{\mathrm{logk}}$= k

7. $(1-\frac{1}{2})+(\frac{1}{3}-\frac{1}{4})+(\frac{1}{5}-\frac{1}{6})\mathrm{........}=\log 2$

or $\frac{1}{1.2}$+$\frac{1}{3.4}$+$\frac{1}{5.6}$+......= log 2.