Find the maclaurin series of f(x) by using known series: f ( x ) = 4 x 8 + x 3

Question

Find the maclaurin series of f(x) by using known series:  f      (    x    )    =            4      x              8      +              x        3

Bruno Hughes · Accepted Answer

I shall use the fact that if, on some interval

(- a, a)

(with

a > 0

) you have

f (x) = \sum_{n = 0}^{\infty} a_{n} x^{n}

, then the Maclaurin series of f is

\sum_{n = 0}^{\infty} a_{n} x^{n}

. In this case, you have

\begin{aligned} \frac{1}{8 + x^{3}} & = \frac{1}{8} \cdot \frac{1}{1 + {(\frac{x}{2})}^{3}} \\ = \frac{1}{8} (1 - {(\frac{x}{2})}^{3} + {(\frac{x}{2})}^{6} - {(\frac{x}{2})}^{9} + \dots) \end{aligned}

if

| x | < 2

. So, still assuming that

| x | < 2

\frac{4 x}{8 + x^{3}} = \frac{1}{2} x - \frac{1}{2^{4}} x^{4} + \frac{1}{2^{7}} x^{7} - \frac{1}{2^{10}} x^{10} + \dots

Find the maclaurin series of f(x) by using known series: f ( x ) </mrow> =