# Chapter 10: Infinite Sequences and Series - Section 10.10 - The Binomial Series and Applications of Taylor Series - Exercises 10.10 - Page 633: 11

$1+4x+6x^2+4x^3+x^4$

#### Work Step by Step

Apply the binomial series formula to determine the first four terms: $(1+x)^r=1+\Sigma_{k=1}^\infty \dbinom{r}{k}x^k$ Here, we have $\dbinom{r}{k}=\dfrac{r(r-1)(r-2).....(r-k+1)}{k!}$ $(1+x)^{4}=1+4x+\dfrac{(4)(3)}{2!}x^2+\dfrac{(4)(3)(2)}{3!}x^3+\dfrac{(4)(3)(2)(1)}x^4{4!}+...$ Hence, our first four terms are: $1+4x+6x^2+4x^3+x^4$

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