Sum to infinity the series 2.3/3!+3.5/4!+4.7/5!+5.9/6!+
Answers
Given : 2.3/3!+3.5/4!+4.7/5!+5.9/6!+
To Find : Sum to infinity the series
Solution:
2.3/3!+3.5/4!+4.7/5!+5.9/6!+
= ∑ (n + 2)(2n +3) / (n + 3)! taking n from 0 to infinity
= ∑(2n² + 7n + 6)/ (n + 3)!
2n² + 7n + 6 = 2n² + 7n + 3 + 3 = 2n² + 6n + n + 3 + 3
= 2n(n + 3) + 1(n + 3) + 3= (n + 3)(2n + 1) + 3
= ∑ (( n + 3)(2n + 1) + 3)/ (n + 3)!
=∑ ( n + 3)(2n + 1)/ (n + 3)! + 3∑1/(n + 3)!
= ∑(2n + 1)/(n + 2)! + 3∑1/(n + 3)!
= ∑(2n + 4 - 3)/(n + 2)! + 3∑1/(n + 3)!
= 2∑(n + 2) /(n + 2)! - 3∑1/(n + 2)! + 3∑1/(n + 3)!
= 2∑1/(n + 1)! - 3( ∑1/(n + 2)! - ∑1/(n + 3)! )
= 2∑1/(n + 1)! - 3 ( 1/2! - 1/3! + 1/3! - 1/4! + 1/4! --- ----- )
= 2∑1/(n + 1)! - 3/ ( 1/2!)
= 2∑1/(n + 1)! - 3/2
∑1/(n + 1)! = e - 1/0! = e - 1
= 2 ( e - 1) - 3/2
= 2e - 7/2
e ≈ 2.7183
= 1.9366
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