Use the nth term test for divergence to determine which, if any, of the following infinite series diverge(s). (10 points) I got D ( OR all of them ), did I do this right?
I. the summation from n equals 1 to infinity of the nth power of the quotient of the quantity n plus 3 and n
II. the summation from n equals 1 to infinity of the product of n and the tangent of pi over n
III. the summation from n equals 1 to infinity of the quotient of the quantity n squared plus 1 and the quantity n squared plus 2
A) I only
B) II only
C) II and III
D) All of them
so, did you do the test?
Clearly not D, since (n+3)/n → 1
To determine whether the given infinite series converge or diverge using the nth term test, we need to examine the behavior of the individual terms as n approaches infinity.
I. The series is the summation from n equals 1 to infinity of (n+3)/n^2.
To apply the nth term test, we need to take the limit as n approaches infinity of the nth term. In this case, the nth term is (n+3)/n^2.
Taking the limit as n approaches infinity:
lim(n->∞) (n+3)/n^2
We can simplify this by dividing each term of the fraction by n^2:
lim(n->∞) (1/n + 3/n^2)
As n approaches infinity, both 1/n and 3/n^2 approach 0. Thus, the limit becomes:
lim(n->∞) (0 + 0) = 0
Since the limit is 0, we cannot definitively determine divergence or convergence using the nth term test. Further analysis is needed.
II. The series is the summation from n equals 1 to infinity of n * tan(π/n).
Again, we will take the limit as n approaches infinity of the nth term. In this case, the nth term is n * tan(π/n).
Taking the limit as n approaches infinity:
lim(n->∞) n * tan(π/n)
As n approaches infinity, π/n goes to 0 and tan(0) = 0. Therefore, the limit becomes:
lim(n->∞) n * 0 = 0
Since the limit is 0, we cannot definitively determine divergence or convergence using the nth term test. Further analysis is needed.
III. The series is the summation from n equals 1 to infinity of (n^2 + 1)/(n^2 + 2).
Once again, we will take the limit as n approaches infinity of the nth term. In this case, the nth term is (n^2 + 1)/(n^2 + 2).
Taking the limit as n approaches infinity:
lim(n->∞) (n^2 + 1)/(n^2 + 2)
Dividing each term by n^2:
lim(n->∞) (1 + 1/n^2)/(1 + 2/n^2)
As n approaches infinity, both 1/n^2 and 2/n^2 approach 0. Thus, the limit becomes:
lim(n->∞) (1 + 0)/(1 + 0) = 1/1 = 1
Since the limit is not equal to 0, the series does not satisfy the nth term test for divergence. Therefore, this series converges.
From the analysis above, we can see that only the third series (III) converges, while the first series (I) and second series (II) do not converge. Therefore, the correct answer is:
A) I only