# Use the Binomial Theorem to find the first five terms of the Maclaurin series f ( x ) = 3 ? 1 + 2 x

## Question:

Use the Binomial Theorem to find the first five terms of the Maclaurin series

$$f(x) = \enspace \sqrt[3]{1+2x}$$

## The Binomial Theorem:

The binomial theorem of the function is used to show the expansion, as the same way we use the Maclaurin series of the Taylor series as the expansion techniques. Now, the series can be converging or diverging.

\

So we will apply the binomial theorem and expand using the formula:

{eq}\left(a+b\right)^n=\sum _{i=0}^n\binom{n}{i}a^{\left(n-i\right)}b^i\\ {/eq}

here we have the function;

{eq}f(x) = \enspace \sqrt[3]{1+2x}\\ \Rightarrow a=1\\ b=2x\\ n=\frac{1}{3}\\ {/eq}

So the expansion upto first five terms will be :

{eq}\enspace \sqrt[3]{1+2x}= \sum _{i=0}^5 \binom{\frac{1}{3}}{i} (2x)^i\\ =1+\frac{2}{3}x-\frac{4}{9}x^2+\frac{40}{81}x^3-\frac{160}{243}x^4 {/eq}