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        "name": "16_3 The Binomial Series",
        "intro": "<p dir=\"ltr\" style=\"text-align: left;\">In this Section we examine an important example of an infinite series, the binomial series:<\/p><p dir=\"ltr\" style=\"text-align: left;\">\\[1+px+\\frac{p(p\u22121)}{2!}x^2+\\frac{p(p\u22121)(p\u22122)}{3!}x^3+\\cdots \\]<\/p><p dir=\"ltr\" style=\"text-align: left;\">We show that this series is only convergent if \\(|x|&lt;1\\)and that in this case the series sums to the value \\((1 +x)^p\\).&nbsp;<\/p><p dir=\"ltr\" style=\"text-align: left;\">As a special case of the binomial series we consider the situation when \\( p\\) is a positive integer \\(n\\). In this case the infinite series reduces to a finite series and we obtain, by replacing \\(x\\) with \\(\\dfrac{a}{b}\\) and multiplying by \\(b^n\\), the usual&nbsp;<strong>binomial theorem<\/strong>:<\/p><p dir=\"ltr\" style=\"text-align: left;\">\\[(b+a)^n=b^n+nb^{n\u22121}a+\\frac{n(n\u22121)}{2!}b^{n\u22122}a^2+\\cdots+a^n.\\]<\/p><p dir=\"ltr\" style=\"text-align: left;\">Finally, we use the binomial series to obtain various polynomial expressions for \\((1 +x)^p\\) when \\(x\\) is \u2018small\u2019.<br><\/p>",
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