Series

1.

Given an analytic function p(V) one can expand that function about the point V=a i.e.,

$$\begin{eqnarray*}p(V) = p(a) + \left . \frac{dp(V)}{dV}\right \vert _a (V-a) + \... ...{2!} \left . \frac{d^2p(V)}{dV^2}\right \vert _a (V-a)^2+ \cdots \end{eqnarray*}$$

where p(a) is the pressure at V=a, $\left . \frac{dp(V)}{dV}\right \vert _a$ is the derivative of p(V) with respect to V evaluated at V=a, i.e., the slope, etc..

2.

Some important series are

(a)

Geometric

(b)

Binomial

There is an error in the following (thanks Mike Daly). It should read 1 + nx + etc..

(c)

Natural Logarithm (ln)

(d)

Exponential

(e)

Sin

$$\sin (x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots (-\infty \leq x \leq +\infty)$$

(f)

Cos

$$\cos (x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots (-\infty \leq x \leq +\infty)$$

Given an analytic function p(V) one can expand that function about the point V=a i.e.,

$$\begin{eqnarray*}p(V) = p(a) + \left . \frac{dp(V)}{dV}\right \vert _a (V-a) + \... ...{2!} \left . \frac{d^2p(V)}{dV^2}\right \vert _a (V-a)^2+ \cdots \end{eqnarray*}$$

where p(a) is the pressure at V=a, $\left . \frac{dp(V)}{dV}\right \vert _a$ is the derivative of p(V) with respect to V evaluated at V=a, i.e., the slope, etc.. Some important series are

1.

Geometric

$$\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots (-1 < x < 1)$$< x < 1) \end{displaymath}">

2.

Binomial

$$(1+x)^n = 1 = nx + \frac{n(n-1)}{2}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 +\cdots (-1 < x < 1 )$$< x < 1 ) \end{displaymath}">

3.

Natural Logarithm

$$\ell n (1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots (-1 < x < 1 )$$< x < 1 ) \end{displaymath}">

4.

Exponential

$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots (-\infty \leq x \leq +\infty)$$

5.

Sin

$$\sin (x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots (-\infty \leq x \leq +\infty)$$

6.

Cos

$$\cos (x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots (-\infty \leq x \leq +\infty)$$