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Given the expression \$$(\\var{a}+\\var{b}x)^{\\var{n}}\$$

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The binomial series expansion for an expression of the form \$$(a+bx)^n\$$ where \$$n\$$ is a Natural number is given by:

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\$$(a+bx)^n=\\tbinom{n}{0}(a)^n(bx)^{0}+\\tbinom{n}{1}(a)^{n-1}(bx)^{1}+\\tbinom{n}{2}(a)^{n-2}(bx)^{2}+...\\tbinom{n}{k}(a)^{n-k}(bx)^{k}+...\\tbinom{n}{n}(a)^{0}(bx)^{n}\$$

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In this example  \$$n=\\var{n}\$$,  \$$a=\\var{a}\$$  and  \$$b=\\var{b}\$$.

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So the first three terms of the binomial series expansion are:

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\$$\\var{a}^{\\var{n}}+\\tbinom{\\var{n}}{\\var{1}}\\times\\var{a}^{\\var{n}-1}\\times\\var{b}^{1}+\\tbinom{\\var{n}}{2}\\times\\var{a}^{\\var{n}-2}\\var{b}^{2}\$$

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\$$=\\simplify{{a}^{n}}+\\simplify{{n}*{a}^({n}-1)*{b}}x+\\simplify{{n}*{n-1}*{a}^{{n}-2}*{b}^2/2x^2}\$$

\n

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Write out the first three terms of the binomial expansion.

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Find the first 3 terms of Binomial series having a Natural exponent

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