欢迎光临散文网会员登陆 & 注册

[Calculus] Infinite Arithmetic-Geometric Series

2021-07-19 20:47 作者:AoiSTZ23 0人读过 | 我要投稿

By: Tao Steven Zheng (郑涛)

【Problem】

This problem is one of my favourite problems recorded in one of my high school mathematical notebooks!

Show that if $%7Cr%7C%20%3C%201$ , and $a%2C%20b$ are constants, then

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

【Solution】

Break the series into two parts

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

When $%7Cr%7C%20%3C%201%20$ , the first part of the sum is a convergent infinite geometric series. The formula for calculating this series is well known:

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

Now consider the second part of the sum

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

Factor our $br$ :

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

It is known that from the convergent infinite geometric series that

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

Notice that by differentiating the left-hand side of the above series, we get

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

Consequently,

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

and

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

Therefore,

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

标签：

[Calculus] Infinite Arithmetic-Geometric Series的评论 (共条)