Question

sum of (n)/(e^(3n)), n=1 to inf

Answer 1

PS: Use the integral test to see if the series converges or diverges.

Answer 2

Σ(n = 1 to ∞) (n+1)(2n+1)/(n+2)!
= Σ(n = 1 to ∞) ((n+2) - 1)(2(n+2) - 3)/(n+2)!
= Σ(n = 1 to ∞) (2(n+2)^2 - 5(n+2) + 3)/(n+2)!
= Σ(n = 1 to ∞) [2(n+2)/(n+1)! - 5/(n+1)! + 3/(n+2)!]
= Σ(n = 1 to ∞) [2((n+1)+1)/(n+1)! - 5/(n+1)! + 3/(n+2)!]
= Σ(n = 1 to ∞) [2/n! + 2/(n+1)! - 5/(n+1)! + 3/(n+2)!]
= 2 * Σ(n = 1 to ∞) 1/n! + Σ(n = 1 to ∞) [-3/(n+1)! + 3/(n+2)!]
= 2(e - 1/0!) + Σ(n = 1 to ∞) [-3/(n+1)! + 3/(n+2)!], since Σ(n = 0 to ∞) 1/n! = e
= (2e - 2) + lim(k→∞) [(-3/2! + 3/3!) + (-3/3! + 3/4!) + ... + (-3/(k+1)! + 3/(k+2)!)]
= (2e - 2) + lim(k→∞) (-3/2! + 3/(k+2)!), since all other terms cancel in pairs
= (2e - 2) + (-3/2! + 0)
= 2e - 7/2.
--------------
I hope this helps!

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? answered 9 months ago


your question is not clear

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Σ(n = 1 to ∞) (n+1)(2n+1)/(n+2)!
= Σ(n = 1 to ∞) ((n+2) - 1)(2(n+2) - 3)/(n+2)!
= Σ(n = 1 to ∞) (2(n+2)^2 - 5(n+2) + 3)/(n+2)!
= Σ(n = 1 to ∞) [2(n+2)/(n+1)! - 5/(n+1)! + 3/(n+2)!]
= Σ(n = 1 to ∞) [2((n+1)+1)/(n+1)! - 5/(n+1)! + 3/(n+2)!]
= Σ(n = 1 to ∞) [2/n! + 2/(n+1)! - 5/(n+1)! + 3/(n+2)!]
= 2 * Σ(n = 1 to ∞) 1/n! + Σ(n = 1 to ∞) [-3/(n+1)! + 3/(n+2)!]
= 2(e - 1/0!) + Σ(n = 1 to ∞) [-3/(n+1)! + 3/(n+2)!], since Σ(n = 0 to ∞) 1/n! = e
= (2e - 2) + lim(k→∞) [(-3/2! + 3/3!) + (-3/3! + 3/4!) + ... + (-3/(k+1)! + 3/(k+2)!)]
= (2e - 2) + lim(k→∞) (-3/2! + 3/(k+2)!), since all other terms cancel in pairs
= (2e - 2) + (-3/2! + 0)
= 2e - 7/2.
--------------
I hope this helps!

2

Comment


.


Other Answers (1)




? answered 9 months ago


your question is not clear

1

Comment


.




Sign In to add your answer
.











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Related Questions

Series Sum(infinity; n=1) series 1/n(n+1)(n+2) converges or diverges?
Sum from 1 to infinity of 1/(n^2+1)?
Find the sum of the series sum (2^(n-1))/(5^(n+1)) n=1 to infinity?
Sum of convergent series: ∑ (n=2 to infinity) 1 / (n^2 -1)?
Here's a fun 1! test this series for convergence sum from n=1 to infinity of (2^1/n - 1)?



Discover Questions

Find an element of the largest order in S_n for n = 1,2,3...10?
Help with area in polar coordinates problem?
Can someone help with a little algebra?
What is after 999 Novengoogolplexion?


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