Copyright

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.

Answer and Explanation:


\

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}


Learn more about this topic:

Loading...
The Binomial Theorem: Defining Expressions

from Algebra II: High School

Chapter 12 / Lesson 7
11K

Related to this Question

Explore our homework questions and answers library