Expand 11+x4 as a power series

Question

rogreenhoxa8 · Accepted Answer

We have given an expression to write in the power series form,

\frac{1}{\sqrt[4]{1 + x}}

To write it in the power series form, we equate it with the general form of power series.

\frac{1}{\sqrt[4]{1 + x}} = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} . . .

(1)
Now we plug x=0, we get,

\frac{1}{\sqrt[4]{1 + x}} = a_{0} + 0

a_{0} = 1

Now on differentiating the equation (1) on both sides,

- \frac{1}{4 (1 + x)^{\frac{5}{4}}} = a_{1} + 2 a_{2} x + 3 a_{3} x^{2} . . .

(2)
Now we again plug x=0 and get,

- \frac{1}{4 (1 + 0)^{\frac{5}{4}}} = a_{1} + 0

a_{1} = - \frac{1}{4}

Now on differentiating the equation (2) on both sides,

\frac{5}{16 (1 + x)^{\frac{9}{4}}} = 2 a_{2} + 6 a_{3} x . . .

(3)
Now we again plug x=0 and get,

\frac{5}{16 (1 + 0)^{\frac{9}{4}}} = 2 a_{2} + 0

a_{2} = \frac{5}{32}

Now we keep doing the same process and keep the terms together,

\frac{1}{\sqrt[4]{1 + x}} = 1 + \sum_{n = 1}^{\infty} (- 1)^{n} \frac{1 \cdot 5 \cdot 9 \cdot 13 \cdot . . . (4 n - 3)}{4^{n} n!} x^{n}

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Expand frac{1}{sqrt[4]{1+x}} as a power series