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一种据strong>电源系列据/strong>是一种采用一般形式的无限系列据/p>
N据/span>=据/span>0.据/span>σ.据/span>∞据/span>一种据/span>N据/span>(据/span>X据/span>-据/span>C据/span>)据/span>N据/span>=据/span>一种据/span>0.据/span>+据/span>一种据/span>1据/span>(据/span>X据/span>-据/span>C据/span>)据/span>+据/span>一种据/span>2据/span>(据/span>X据/span>-据/span>C据/span>)据/span>2据/span>+据/span>一种据/span>3.据/span>(据/span>X据/span>-据/span>C据/span>)据/span>3.据/span>+据/span>⋯据/span>那据/span>
在哪里据span class="katex">
一种据/span>N据/span>是一个独立的功能据span class="katex">
X据/span>和据span class="katex">
C据/span>是一个任意常数。因此,每个系数据span class="katex">
(据/span>例如据span class="katex">
一种据/span>1据/span>那据/span>一种据/span>2据/span>那据/span>......据/span>)据/span>可以是结果的结果据span class="katex">
N据/span>TH.据/span>术语被取代成给定公式。假设这一点据span class="katex">
N据/span>TH.据/span>术语据span class="katex">
一种据/span>N据/span>等于据span class="katex">
N据/span>+据/span>1据/span>N据/span>!!据/span>。然后是据span class="katex">
5.据/span>TH.据/span>术语是据span class="katex">
5.据/span>+据/span>1据/span>5.据/span>!!据/span>=据/span>6.据/span>1据/span>2据/span>0.据/span>=据/span>2据/span>0.据/span>。据/p>
电源系列与主题密切相关据一种href="//www.parkandroid.com/wiki/taylor-series/" class="wiki_link" title="" target="_blank">泰勒系列据/strong>因为它可用于近似有关该值的某些功能据span class="katex">
X据/span>=据/span>C据/span>。据/p>
任何电源系列都可以呈现近似值据em>中央据/em>该系列,由常数表示据span class="katex">
C据/span>以上。提供动力系列将收敛,只要它与这一中心的迷失太远。据/p>
电力系列的常见问题涉及找到据em>收敛半径据/em>和据em>收敛间隔据/em>一个系列。据/p>
电力系列最常见的例子之一是据一种href="//www.parkandroid.com/wiki/geometric-progressions/" class="wiki_link" title="" target="_blank">几何系列据/strong>。在这种情况下,价值据span class="katex">
一种据/span>N据/span>始终保持不变,值据span class="katex">
C据/span>设置为0.因此,据/p>
N据/span>=据/span>0.据/span>σ.据/span>∞据/span>一种据/span>X据/span>N据/span>=据/span>一种据/span>+据/span>一种据/span>X据/span>+据/span>一种据/span>X据/span>2据/span>+据/span>一种据/span>X据/span>3.据/span>+据/span>⋯据/span>那据/span> 每个连续的术语都会增加一个因素据span class="katex">
X据/span>。据/p>
电力系列的另一个常见应用是据一种href="//www.parkandroid.com/wiki/maclaurin-series/" class="wiki_link" title="" target="_blank">Maclaurin系列据/strong>,这是一种特殊类型的泰勒系列,附加规定该中心专门涉及该价值据span class="katex">
X据/span>=据/span>0.据/span>。所以,据/p>
F据/span>(据/span>0.据/span>)据/span>≈据/span>N据/span>=据/span>0.据/span>σ.据/span>∞据/span>N据/span>!!据/span>F据/span>(据/span>N据/span>)据/span>(据/span>0.据/span>)据/span>X据/span>N据/span>=据/span>F据/span>(据/span>0.据/span>)据/span>+据/span>F据/span>(据/span>1据/span>)据/span>(据/span>0.据/span>)据/span>X据/span>+据/span>2据/span>!!据/span>F据/span>(据/span>2据/span>)据/span>(据/span>0.据/span>)据/span>X据/span>2据/span>+据/span>3.据/span>!!据/span>F据/span>(据/span>3.据/span>)据/span>(据/span>0.据/span>)据/span>X据/span>3.据/span>+据/span>⋯据/span> 可用于近似任何功能据span class="katex">
F据/span>(据/span>X据/span>)据/span>关于价值据span class="katex">
X据/span>=据/span>0.据/span>当更多术语是求和的一部分时,结果更准确。据/p>
一般来说,表格的函数据span class="katex"> N据/span>=据/span>0.据/span>σ.据/span>∞据/span>F据/span>(据/span>N据/span>)据/span>将收敛据span class="katex"> L.据/span>据据/span>1据/span>这样据/p>
L.据/span>=据/span>N据/span>→据/span>∞据/span>林据/span>|据/span>|据/span>|据/span>|据/span>F据/span>(据/span>N据/span>)据/span>F据/span>(据/span>N据/span>+据/span>1据/span>)据/span>|据/span>|据/span>|据/span>|据/span>。据/span>
这是一个标准过程,被称为据一种href="//www.parkandroid.com/wiki/convergence-ratio-test/" class="wiki_link" title="" target="_blank">比率测试据/strong>。据/p>
找到收敛半径和功率系列的收敛间隔据span class="katex">
N据/span>=据/span>0.据/span>σ.据/span>∞据/span>N据/span>(据/span>X据/span>-据/span>3.据/span>)据/span>N据/span>。据/span> 首先,对功能进行比率测试。据/p>
L.据/span>=据/span>N据/span>→据/span>∞据/span>林据/span>|据/span>|据/span>|据/span>|据/span>F据/span>(据/span>N据/span>)据/span>F据/span>(据/span>N据/span>+据/span>1据/span>)据/span>|据/span>|据/span>|据/span>|据/span>=据/span>N据/span>→据/span>∞据/span>林据/span>|据/span>|据/span>|据/span>|据/span>N据/span>+据/span>1据/span>(据/span>X据/span>-据/span>3.据/span>)据/span>N据/span>+据/span>1据/span>÷据/span>N据/span>(据/span>X据/span>-据/span>3.据/span>)据/span>N据/span>|据/span>|据/span>|据/span>|据/span>=据/span>N据/span>→据/span>∞据/span>林据/span>|据/span>|据/span>|据/span>|据/span>N据/span>+据/span>1据/span>N据/span>(据/span>X据/span>-据/span>3.据/span>)据/span>|据/span>|据/span>|据/span>|据/span>。据/span> 为了隔离功能据span class="katex">
X据/span>,我们可以简单地将其从极限中删除,因为它独立于据span class="katex">
N据/span>:据/p>
L.据/span>=据/span>|据/span>X据/span>-据/span>3.据/span>|据/span>×据/span>N据/span>→据/span>∞据/span>林据/span>|据/span>|据/span>|据/span>|据/span>N据/span>+据/span>1据/span>N据/span>|据/span>|据/span>|据/span>|据/span>=据/span>|据/span>X据/span>-据/span>3.据/span>|据/span>×据/span>1据/span>=据/span>|据/span>X据/span>-据/span>3.据/span>|据/span>。据/span> 鉴于据span class="katex">
L.据/span>据据/span>1据/span>为了使系列融合,它必须是那个据span class="katex">
|据/span>X据/span>-据/span>3.据/span>|据/span>据据/span>1据/span>当它收敛。当你表达表单时据span class="katex">
|据/span>X据/span>+据/span>一种据/span>|据/span>据据/span>R.据/span>和据span class="katex">
一种据/span>∈据/span>R.据/span>, 这据strong>收敛半径据/strong>是值的价值据span class="katex">
R.据/span>。因此,示例中的收敛半径为1。据/p>
这据strong>收敛间隔据/strong>另一方面,所有值的集合都是如此据span class="katex">
X据/span>系列融合为此。使用上述不等式,一定是据/p>
|据/span>X据/span>-据/span>3.据/span>|据/span>-据/span>1据/span>据据/span>X据/span>-据/span>3.据/span>2据/span>据据/span>X据/span>据据/span>1据/span>据据/span>1据/span>据据/span>4.据/span>。据/span> 然而,它并不像它乍一看那么简单。我们还需要检查间隔的边界值以检查该系列是否会聚这些值。因此对于据span class="katex">
X据/span>=据/span>2据/span>那据/span>我们有据/p>
N据/span>=据/span>0.据/span>σ.据/span>∞据/span>N据/span>(据/span>2据/span>-据/span>3.据/span>)据/span>N据/span>=据/span>N据/span>=据/span>0.据/span>σ.据/span>∞据/span>N据/span>(据/span>-据/span>1据/span>)据/span>N据/span>那据/span> 哪个,使用据em>交替系列测试据/em>,收敛。现在,为据span class="katex">
X据/span>=据/span>4.据/span>那据/span>
N据/span>=据/span>0.据/span>σ.据/span>∞据/span>N据/span>(据/span>4.据/span>-据/span>3.据/span>)据/span>N据/span>=据/span>N据/span>=据/span>0.据/span>σ.据/span>∞据/span>N据/span>1据/span>N据/span>=据/span>N据/span>=据/span>0.据/span>σ.据/span>∞据/span>N据/span>1据/span>那据/span> 作为标准结果,不会收敛。因此,收敛的间隔据span class="katex">
N据/span>=据/span>0.据/span>σ.据/span>∞据/span>N据/span>(据/span>X据/span>-据/span>3.据/span>)据/span>N据/span>是据strong>
2据/span>≤.据/span>X据/span>据据/span>4.据/span>
。据span class="katex">
□据/span>
找到融合半径据span class="katex"> N据/span>=据/span>0.据/span>σ.据/span>∞据/span>N据/span>!!据/span>⋅据/span>X据/span>N据/span>。据/span>
执行函数的比率测试:据/p>
L.据/span>=据/span>N据/span>→据/span>∞据/span>林据/span>|据/span>|据/span>|据/span>|据/span>N据/span>!!据/span>⋅据/span>X据/span>N据/span>(据/span>N据/span>+据/span>1据/span>)据/span>!!据/span>⋅据/span>X据/span>N据/span>+据/span>1据/span>|据/span>|据/span>|据/span>|据/span>=据/span>N据/span>→据/span>∞据/span>林据/span>|据/span>|据/span>X据/span>(据/span>N据/span>+据/span>1据/span>)据/span>|据/span>|据/span>=据/span>|据/span>X据/span>|据/span>⋅据/span>N据/span>→据/span>∞据/span>林据/span>|据/span>|据/span>(据/span>N据/span>+据/span>1据/span>)据/span>|据/span>|据/span>。据/span>
此示例从上面的示例变化,因为限制不会收敛到1据span class="katex"> N据/span>→据/span>∞据/span>。对于任何非零值据span class="katex"> X据/span>那据span class="katex"> L.据/span>将永远等于据span class="katex"> ∞据/span>。唯一的价值据span class="katex"> X据/span>可以采取不同的不同据span class="katex"> 0.据/span>。当这是真的,据span class="katex"> L.据/span>=据/span>0.据/span>那据/span>这也满足了收敛的条件据span class="katex"> (据/span>L.据/span>据据/span>1据/span>)据/span>。据/span>因此,系列的收敛半径为0,收敛的间隔是据span class="katex"> X据/span>=据/span>0.据/span>。据span class="katex"> □据/span>
找到融合半径据span class="katex"> N据/span>=据/span>0.据/span>σ.据/span>∞据/span>2据/span>2据/span>N据/span>(据/span>N据/span>!!据/span>)据/span>2据/span>(据/span>-据/span>1据/span>)据/span>N据/span>X据/span>2据/span>N据/span>。据/span>
执行函数的比率测试:据/p>
L.据/span>=据/span>N据/span>→据/span>∞据/span>林据/span>|据/span>|据/span>|据/span>|据/span>|据/span>2据/span>2据/span>N据/span>+据/span>2据/span>(据/span>(据/span>N据/span>+据/span>1据/span>)据/span>!!据/span>)据/span>2据/span>(据/span>-据/span>1据/span>)据/span>N据/span>+据/span>1据/span>X据/span>2据/span>N据/span>+据/span>2据/span>÷据/span>2据/span>2据/span>N据/span>(据/span>N据/span>!!据/span>)据/span>2据/span>(据/span>-据/span>1据/span>)据/span>N据/span>X据/span>2据/span>N据/span>|据/span>|据/span>|据/span>|据/span>|据/span>=据/span>N据/span>→据/span>∞据/span>林据/span>|据/span>|据/span>|据/span>|据/span>-据/span>4.据/span>(据/span>N据/span>+据/span>1据/span>)据/span>2据/span>X据/span>2据/span>|据/span>|据/span>|据/span>|据/span>=据/span>N据/span>→据/span>∞据/span>林据/span>|据/span>|据/span>|据/span>|据/span>4.据/span>(据/span>N据/span>+据/span>1据/span>)据/span>2据/span>X据/span>2据/span>|据/span>|据/span>|据/span>|据/span>=据/span>|据/span>|据/span>X据/span>2据/span>|据/span>|据/span>⋅据/span>N据/span>→据/span>∞据/span>林据/span>|据/span>|据/span>|据/span>|据/span>4.据/span>(据/span>N据/span>+据/span>1据/span>)据/span>2据/span>1据/span>|据/span>|据/span>|据/span>|据/span>。据/span>
此方案中的限制方法0,因此此结果将适用于所有据span class="katex"> X据/span>。这意味着据span class="katex"> X据/span>可以带来任何价值和据span class="katex"> L.据/span>=据/span>0.据/span>据据/span>1据/span>那据/span>意思就是据span class="katex"> N据/span>=据/span>0.据/span>σ.据/span>∞据/span>2据/span>2据/span>N据/span>(据/span>N据/span>!!据/span>)据/span>2据/span>(据/span>-据/span>1据/span>)据/span>N据/span>X据/span>2据/span>N据/span>将为所有人融合据span class="katex"> X据/span>。因此,在这种情况下收敛半径是据span class="katex"> ∞据/span>并且收敛的间隔是据span class="katex"> -据/span>∞据/span>据据/span>X据/span>据据/span>∞据/span>。据span class="katex"> □据/span>
以下电力系列的收敛间隔是多少;也就是说,对于什么价值据span class="katex"> X据/span>该系列是否会聚?据/p>
N据/span>=据/span>0.据/span>σ.据/span>∞据/span>4.据/span>N据/span>(据/span>-据/span>1据/span>)据/span>N据/span>N据/span>(据/span>X据/span>+据/span>3.据/span>)据/span>N据/span>
示例1和2从Calculus 7e提前超越James Stewart。据/p>
特别感谢托马斯·赫尔顿学院的数学教授。据/p>
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