Uji kekonvergenan: Perbedaan antara revisi
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===[[:en:Integral test for convergence|Tes integral]]===
Deret itu dapat dibandingkan dengan suatu integral untuk menguji apakah konvergen atau divergen.
Misalnya <math>f:[1,\infty)\to\R_+</math> :Jika <math>\int_{1}^{\infty} f(x)\, dx = \lim_{t \to \infty} \int_{1}^{t} f(x)\, dx < \infty,</math> maka deret itu konvergen
:Jika integral itu divergen, maka deret itu juga divergen.
===[[Direct comparison test]]===
===[[Limit comparison test]]===
'''
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===[[Cauchy condensation test]]===
Let <math>\left \{ a_n \right \}</math> be a positive non-increasing sequence. Then the sum <math>A = \sum_{n=1}^\infty a_n</math> converges [[if and only if]] the sum <math>A^* = \sum_{n=0}^\infty 2^n a_{2^n}</math> converges. Moreover, if they converge, then <math>A \leq A^* \leq 2A</math> holds.
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Suppose the following statements are true:
Baris 61 ⟶ 62:
This is also known as the Leibniz criterion. If <math>\sum_{n=1}^\infty a_n</math> is a series whose terms alternative from positive to negative, and if the limit as n approaches infinity of <math> a_n </math> is zero and the absolute value of each term is less than the absolute value of the previous term, then <math>\sum_{n=1}^\infty a_n</math> is convergent.
===[[
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▲===[[Raabe-Duhamel's test]]===
Definisikan
▲Let { ''a''<sub>n</sub> } > 0.
<math> b_n = n \left( \frac{ a_n }{ a_{ n + 1 } } - 1 \right ) </math>.
Jika <math> L = \lim_{ n \to \infty } b_n </math> ada, maka ada tiga kemungkinan:▼
▲<math> L = \lim_{ n \to \infty } b_n </math>
* Jika ''L'' > 1 deret itu konvergen
* Jika ''L'' < 1 deret itu divergen
* Jika ''L'' = 1 tes itu tidak konklusif.
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An alternative formulation of this test is as follows. Let { ''a''<sub>n</sub> } be a series of real numbers. Then if ''b'' > 1 and K (a natural number) exist such that
Baris 87 ⟶ 84:
for all ''n'' > ''K'' then the series { ''a''<sub>n</sub> } is convergent.
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*For some specific types of series there are more specialized convergence tests, for instance for [[Fourier series]] there is the [[Dini test]]
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convergence follows from the root test but not from the ratio test.
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Consider the series
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(**) is geometric series with ratio <math> 2^{(1-\alpha)} </math>. (**) is finitely convergent if its ratio is less than one (namely <math>\alpha > 1</math>). Thus, (*) is finitely convergent [[if and only if]] <math> \alpha > 1 </math>.
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While most of the tests deal with the convergence of infinite series, they can also be used to show the convergence or divergence of infinite products. This can be achieved using following theorem: Let <math>\left \{ a_n \right \}_{n=1}^\infty</math> be a sequence of positive numbers. Then the infinite product <math>\prod_{n=1}^\infty (1 + a_n)</math> converges [[if and only if]] the series <math>\sum_{n=1}^\infty a_n</math> converges. Also similarly, if <math>0 < a_n < 1</math> holds, then <math>\prod_{n=1}^\infty (1 - a_n)</math> approaches a non-zero limit if and only if the series <math>\sum_{n=1}^\infty a_n</math> converges .
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== Lihat pula ==
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