infinite series

Question

find he sum of the series of it converges else show that it diverges

∞

∑((1/n^2+7n+12)+((2^2n+1)/(5^n))

n->∞

Answer 1

I am a bit worried about precedence of operations issues in you question. Also, you do not make it clear whether the sum starts at n = 0 or n =1

So I will address the case that n starts at n = 0 and the required sum is the sum of two terms:

The first one (A) is

A = Σ_n=0 ^∞ 1 /( n² + 7 n + 12) and the second one (B) is B = Σ_n=0^∞ 2²ⁿ⁺¹ / 5ⁿ

For A, the expression 1/(n² + 7n +12) can be rewritten as 1/(n+3) - 1/(n+4) . By writing out the first few terms it can be seen that this is a telescoping series.

A = 1/3 - 1/4 + 1/4 - 1/5 + 1/5 - 1/6 +1/6 and so on. Thus A = 1/3

For B, the expression 2²ⁿ⁺¹ / 5ⁿ = 2 4ⁿ / 5ⁿ = 2 (4/5)ⁿ This is a geometric series with the common ratio = 4/5

So B = 2 /(1- (4/5) ) = 10

A + B = 10 + 1/3