分数二项式定理据/h1>
已经有一个帐户?据一种href="//www.parkandroid.com/account/login/?next=/wiki/fractional-binomial-theorem/" class="ax-click" data-ax-id="clicked_signup_modal_login" data-ax-type="link">这里登录。据/一种>据/P.>据/div>
这据一种href="//www.parkandroid.com/wiki/binomial-theorem-n-choose-k/" class="wiki_link" title="二项式定理" target="_blank">二项式定理据/一种>为整数指数可以推广到分数指数。相关的麦克劳林系列带来一些有趣的身份(包括据一种href="//www.parkandroid.com/wiki/generating-functions-solving-recurrence-relations/" class="wiki_link" title="生成函数" target="_blank">生成函数据/一种>)和微积分等应用。据/P.>据P.>例如,据span class="katex">
F据/span>(据/span>X据/span>)据/span>=据/span>1据/span>+据/span>X据/span>
=据/span>(据/span>1据/span>+据/span>X据/span>)据/span>1据/span>/据/span>2据/span>不是一个多项式。虽然积极的整数次幂据span class="katex">
1据/span>+据/span>X据/span>可扩展成多项式据span class="katex">
(据/span>例如据span class="katex">
(据/span>1据/span>+据/span>X据/span>)据/span>3.据/span>=据/span>1据/span>+据/span>3.据/span>X据/span>+据/span>3.据/span>X据/span>2据/span>+据/span>X据/span>3.据/span>)据/span>那据/span>这并没有使据span class="katex">
F据/span>(据/span>X据/span>)据/span>多项式,所以不能有单项条款,等于的有限和据span class="katex">
F据/span>(据/span>X据/span>)据/span>。但有恢复同类型的扩展,如果无限资金被允许的方式。据/P.>据P.>作为第一个近似值,因为据span class="katex">
F据/span>'据/span>(据/span>0.据/span>)据/span>=据/span>2据/span>1据/span>的切线据span class="katex">
1据/span>+据/span>X据/span>
在据span class="katex">
X据/span>=据/span>0.据/span>是据span class="katex">
y据/span>=据/span>1据/span>+据/span>2据/span>1据/span>X据/span>。因此,对于小据span class="katex">
X据/span>那据/span>
1据/span>+据/span>X据/span>
≈据/span>1据/span>+据/span>2据/span>1据/span>X据/span>。据/span>这种近似已经是相当有用的,但它可以更仔细地逼近这个功能使用系列。据/P.>据B.lockquote class="example">
扩张据span class="katex">
1据/span>+据/span>X据/span>
作为麦克劳林系列。据/P.>据hr>
这据一种href="//www.parkandroid.com/wiki/power-rule/" class="wiki_link" title="电源规则" target="_blank">电源规则据/一种>在演算可以推广到使用分数指数据一种href="//www.parkandroid.com/wiki/chain-rule/" class="wiki_link" title="链式法则" target="_blank">链式法则据/一种>:该衍生物的据span class="katex">
X据/span>P.据/span>/据/span>问:据/span>是据span class="katex">
问:据/span>P.据/span>X据/span>P.据/span>/据/span>问:据/span>-据/span>1据/span>。据/P.>据P.>现在,让我们据span class="katex">
F据/span>(据/span>X据/span>)据/span>=据/span>1据/span>+据/span>X据/span>
。据/span>然后据/P.>据P.>据span class="katex-display">
F据/span>(据/span>X据/span>)据/span>=据/span>(据/span>1据/span>+据/span>X据/span>)据/span>1据/span>/据/span>2据/span>F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>2据/span>1据/span>(据/span>1据/span>+据/span>X据/span>)据/span>-据/span>1据/span>/据/span>2据/span>F据/span>'据/span>'据/span>(据/span>X据/span>)据/span>=据/span>2据/span>1据/span>⋅据/span>(据/span>-据/span>2据/span>1据/span>(据/span>1据/span>+据/span>X据/span>)据/span>-据/span>3.据/span>/据/span>2据/span>)据/span>F据/span>(据/span>K.据/span>)据/span>(据/span>X据/span>)据/span>=据/span>2据/span>1据/span>⋅据/span>(据/span>-据/span>2据/span>1据/span>)据/span>⋯据/span>(据/span>-据/span>2据/span>2据/span>K.据/span>-据/span>3.据/span>(据/span>1据/span>+据/span>X据/span>)据/span>-据/span>(据/span>2据/span>K.据/span>-据/span>1据/span>)据/span>/据/span>2据/span>)据/span>⇒据/span>F据/span>(据/span>0.据/span>)据/span>=据/span>1据/span>⇒据/span>F据/span>'据/span>(据/span>0.据/span>)据/span>=据/span>2据/span>1据/span>⇒据/span>F据/span>'据/span>'据/span>(据/span>0.据/span>)据/span>=据/span>2据/span>1据/span>⋅据/span>-据/span>2据/span>1据/span>⋮据/span>⇒据/span>F据/span>