散文網(wǎng) » 科技 »學(xué)習(xí) » 【趣味數(shù)學(xué)題】無窮算術(shù)-幾何級(jí)數(shù)

【趣味數(shù)學(xué)題】無窮算術(shù)-幾何級(jí)數(shù)

2021-07-19 20:47 作者:AoiSTZ23 0人讀過 | 我要投稿

?鄭濤 (Tao Steven Zheng) 著

【問題】

這道題是我高中時(shí)最喜歡的數(shù)學(xué)問題之一！

證明如果 $%7Cr%7C%20%3C%201$ , $a%E3%80%81b$ 都是常數(shù), 計(jì)算無窮算術(shù)-幾何級(jí)數(shù) [1]（infinite arithmetic-geometric series）是

$%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%20(a%2Bnb)%7Br%7D%5E%7Bn%7D%20%3D%20%5Cfrac%7Ba%7D%7B1-r%7D%20%2B%20%5Cfrac%7Bbr%7D%7B(1-r)%5E2%7D$

[1] 算術(shù)級(jí)數(shù)（arithmetic series）也叫 "等差級(jí)數(shù)"

【題解】

把級(jí)數(shù)分成兩部分

$%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%20(a%2Bnb)%7Br%7D%5E%7Bn%7D%20%3D%20%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%20a%7Br%7D%5E%7Bn%7D%20%2B%20%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%20nb%7Br%7D%5E%7Bn%7D$

如果 $%7Cr%7C%20%3C%201%20$ , 第一部分是收斂的無窮幾何級(jí)數(shù)（geometric series, 也叫 "等比級(jí)數(shù)"）。計(jì)算這個(gè)幾何級(jí)數(shù)的公式是

$%20%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%20a%7Br%7D%5E%7Bn%7D%20%3D%20a(1%2Br%2Br%5E2%2B...)%20%3D%20%5Cfrac%7Ba%7D%7B1-r%7D$

現(xiàn)在考慮和的第二部分

$%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%20nb%7Br%7D%5E%7Bn%7D%20%3D%20br%2B2br%5E2%2B3br%5E3%2B...$

然后分解出 $br$

$%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%20nb%7Br%7D%5E%7Bn%7D%20%3D%20br(1%2B2r%2B3r%5E2%2B...)$

收斂的無窮幾何級(jí)數(shù)是

$1%2Br%2Br%5E2...%20%3D%20%5Cfrac%7B1%7D%7B1-r%7D$

通過求無窮幾何級(jí)數(shù)的導(dǎo)函數(shù)（derivative）, 得

$1%2B2r%2B3r%5E2%2B...%20%3D%20%5Cfrac%7Bd%7D%7Bdr%7D%5Cleft(%5Cfrac%7B1%7D%7B1-r%7D%5Cright)$

$1%2B2r%2B3r%5E2%2B...%20%3D%20%5Cfrac%7B1%7D%7B(1-r)%5E2%7D$

所以,

$%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%20nb%7Br%7D%5E%7Bn%7D%20%3D%20%5Cfrac%7Bbr%7D%7B(1-r)%5E2%7D$

因此,

$%20%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%20(a%2Bnb)%7Br%7D%5E%7Bn%7D%20%3D%20%5Cfrac%7Ba%7D%7B1-r%7D%20%2B%20%5Cfrac%7Bbr%7D%7B(1-r)%5E2%7D%20$

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