Welcome 2016! Part 36

Algebra Level 3

Evaluate ∑ n = 1 ∞ 3 n 2 + 3 n + 2 . \text{Evaluate } \quad \large \sum_{n = 1}^{\infty} \frac{3}{n^2 + 3n + 2}.


The answer is 1.5.

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Anish Harsha by Anish Harsha , 20, India

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1 solution

Rishabh Jain
Jan 10, 2016

S can be written as 3 ∑ r = 1 ∞ 1 r 2 + 3 r + 2 = 3 ∑ r = 1 ∞ 1 ( r + 2 ) ( r + 1 ) 3\displaystyle \sum_{r=1}^∞ \frac{1}{r^2+3r+2}=3\displaystyle \sum_{r=1}^∞ \frac{1}{(r+2)(r+1)} 3 ∑ r = 1 ∞ 1 r + 1 − 1 r + 2 ( T e l e s c o p i c s e r i e s ) 3\displaystyle \sum_{r=1}^∞ \frac{1}{r+1}-\frac{1}{r+2} \space\space \color{#D61F06}{(Telescopic\space series)} = 3 ( 1 1 + 1 ) = 3 / 2 = 1.5 =3(\frac{1}{1+1})\Large\color{magenta}{=3/2=1.5}

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