Generalize it!

Algebra Level 3

∑ n = 1 ∞ 1 ( 5 + n ) ( 4 + n ) = ? \large \displaystyle\sum _{ n=1 }^{ \infty }{ \frac { 1 }{ (5+n)(4+n) } } = \, ?


The answer is 0.2.

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Hamza A by Hamza A , 19, USA

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2 solutions

Mateus Gomes
Feb 13, 2016

∑ n = 1 ∞ 1 ( 4 + n ) ( 5 + n ) = ∑ n = 1 ∞ ( 1 4 + n − 1 5 + n ) ( T e l e s c o p i n g s e r i e ) = 1 5 = 0.2 \sum_{n=1}^{\infty}\frac{1}{(4+n)(5+n)}=\sum_{n=1}^{\infty}\left(\frac{1}{4+n}-\frac{1}{5+n}\right)~~(\color{#D61F06}{Telescoping~~ serie})=\Large\color{#3D99F6}{\boxed{\frac{1}{5}=0.2}}

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in general,

∑ n = 1 ∞ 1 ( a + n ) ( a + n − 1 ) = 1 a \displaystyle\sum _{ n=1 }^{ \infty }{ \frac { 1 }{ (a+n)(a+n-1) } } =\frac{1}{a}

generalizing stuff is always cool :)

Hamza A - 5 years, 4 months ago
Aman Rckstar
Feb 16, 2016

Too Easy , Just solve using wiki.

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