Find the third degree Taylor polynomial for f(x)=ln⁡x, centered at a=2?

Question

trisanualb6 · Accepted Answer

The general form of a Taylor expansion centered at a of an analytical function f:

f (x) = \sum_{n = 0}^{\infty} \frac{f^{n} (a)}{n!} {(x - a)}^{n}

.

f^{n}

is the n-th derivative of

f

The third degree Taylor polynomial consists of the first four (n ranging from 0 to 3) terms of the full Taylor expansion.

f (a) + f^{'} (a) (x - a) + \frac{f^{″} (a)}{2} {(x - a)}^{2} + \frac{f^{‴} (a)}{6} {(x - a)}^{3}

f (x) = \ln (x)

,

∴ f^{'} (x) = \frac{1}{x}, f^{″} (x) = \frac{1}{x^{2}}, f^{‴} (x) = \frac{2}{x^{3}}

. Thus, the thirs Taylors polynomial is:

\ln (a) + \frac{1}{a} (x - a) - \frac{1}{2 a^{2}} {(x - a)}^{2} + \frac{1}{3 a^{3}} {(x - a)}^{3}

If we have

a = 2

, then

\ln (2) + \frac{1}{2} (x - 2) - \frac{1}{8} {(x - 2)}^{2} + \frac{1}{24} {(x - 2)}^{3}

Find the third degree Taylor polynomial for f(x)=\ln x, centered