Infinite series question.
an=2+(0.99)^n
Converge or Diverge?
look at .99^n
.99^5 = .95099049.....
.99^250 = .08105...
as n ---> ∞ the value of .99^n approaches zero
so you are left with
2 + 0
so it converges to the value of 2
To determine whether the given series converges or diverges, we need to examine the behavior of the terms as n approaches infinity.
The given series is expressed as an infinite sum:
∑ (2 + (0.99)^n)
For a series to converge, the limit of the terms as n approaches infinity must approach zero. Let's calculate the limit of the terms:
lim (n→∞) (2 + (0.99)^n)
To evaluate this limit, we consider that (0.99)^n approaches zero as n approaches infinity. Therefore:
lim (n→∞) (2 + (0.99)^n) = 2 + 0 = 2
Since the limit of the terms is a non-zero value, the series does not converge. Therefore, the given series diverges.