Find if the sequence an = 5/(n+2) diverges or converges.
I think it diverges because the limit as x approaches infinity does not exist.
you know that 1/n diverges
so, also does 1/(n+2) since
∞
∑ 1/(n+2)
n=1
is the same as
∞
∑ 1/n
n=3
lopping off the first two terms and multiplying by 5 doesn't affect the divergence
To determine if the sequence {an} = 5/(n+2) converges or diverges, we need to analyze the behavior of the terms as n approaches infinity.
First, let's compute some terms of the sequence:
a₁ = 5/(1+2) = 5/3 ≈ 1.67
a₂ = 5/(2+2) = 5/4 = 1.25
a₃ = 5/(3+2) = 5/5 = 1
a₄ = 5/(4+2) = 5/6 ≈ 0.83
...
We can observe that as n increases, the terms of the sequence {an} approach zero. However, this is not enough to conclude that the sequence converges.
To make a definitive determination, we need to take the limit as n approaches infinity. Let's calculate the limit:
lim(n→∞) 5/(n+2)
To evaluate this limit, we consider the highest power of n in the denominator. In this case, it is n.
Dividing both the numerator and denominator by n, we get:
lim(n→∞) (5/n)/(1+2/n)
As n approaches infinity, both 5/n and 2/n tend to zero. Therefore, we can simplify the expression further:
lim(n→∞) 0/(1+0) = 0/1 = 0
Since the limit of the terms of the sequence is zero, we can conclude that the sequence {an} = 5/(n+2) converges.
Hence, your initial assumption that it diverges was incorrect.