## Exponential Series

### Exponential Series

**Definition (The number e)**

The limiting value of \({ \left( 1+\frac { 1 }{ n } \right) }^{ n }\) when n tends to infinity is denoted by e.

### Properties of e

(1) e lies between 2.7 and 2.8. i.e., 2.7 < e < 2.8.

(2) The value of e correct to 10 places of decimals is 2.7182818284.

(3) e is an irrational (incommensurable) number.

(4) e is the base of natural logarithm (Napier logarithm) i.e., ln x = log_{e} x and log_{10} e is known as Napierian constant. log_{10} e = 0.43429448, ln x = 2.303 log_{10} x

### Expansion of exponential series

The above series known as exponential series and is called exponential function. Exponential function is also denoted by exp. i.e., e^{x } exp A = e^{A}; ∴ exp x = e^{x}.

Replacing x by -x, we obtain

### Exponential function a^{x}, where a > 0

### Some standard results from exponential series