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Hello!

I have been trying to play around with this summation for a while, but was stuck at a certain place.

 

Problem: \(\sum_{n = 1}^\infty \frac{2n + 1}{n(n + 1)(n + 2)}.\)

 

My intake was to use the "Partial Fraction Decomposition method"

 

\(\frac{2n+1}{n(n+1)(n+2)}=\frac{A}{n}+\frac{B}{n+1}+\frac{C}{n+2}\).

 

After some expanding, evaluating, and simplifying, I ended up with \(A=\frac{1}{2}, B=1, \) and \(C=\frac{-3}{2}\).

 

Therefore, I rewrote the series as: \(\sum_{n = 1}^\infty \frac{2n + 1}{n(n + 1)(n + 2)}=\sum_{n = 1}^\infty \frac{\frac{1}{2}}{n}+\frac{1}{n+1}+\frac{\frac{-3}{2}}{n+2}.\).

 

After this, I'm stuck, and I wonder if this series would telescope..Any help would be appreciated.

 

 

Thank you !

 Mar 30, 2020
 #1
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+1

∑[(2n+1) / (n*(n+1)*(n+2)), n, 1, ∞ ] = It converges to 1.25

 

Here is the Patial Sum Formula:

 

sum_(n=1)^m (2 n + 1)/(n (n + 1) (n + 2)) = (5 m^2 + 7 m)/(4 (m + 1) (m + 2))

 Mar 30, 2020

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