Select the true statement for the series the summation from n=1 to infinity of n!/(2n-1)
a) The series converges by the ratio test.
b) The series diverges by the integral test.
c) The series converges by the integral test.
d) The series diverges by the nth term test.
My answer is b or d?. Can you ckeck for me please. thanks
I like D
clearly, for sufficiently large n, n! > 2n-1 so the nth term does not approach zero
To determine whether the series the summation from n=1 to infinity of n!/(2n-1) converges or diverges, we can use different tests. Let's go through each option and find the correct one:
a) The ratio test: The ratio test states that if the limit as n approaches infinity of the absolute value of (a_{n+1}/a_n) is less than 1, then the series converges. In this case, let's apply the ratio test to the series.
lim (n->∞) |((n+1)!/(2(n+1)-1))/(n!/(2n-1))|
= lim (n->∞) |(n+1)!/(2n+1)(2n-1)| * |(2n-1)/(2n+1)|
= lim (n->∞) |(n+1)/(2n+1)|
= 1/2
The limit is not less than 1, so the ratio test is inconclusive for this series.
b) The integral test: The integral test states that if the function f(x) is positive, continuous, and decreasing for x >= N (where N is some positive integer), and the series a_n = f(n), then the series and the corresponding improper integral ∫f(x)dx both converge or both diverge.
Let's find the integral of the function f(x) = x!/(2x-1):
∫ (x!/(2x-1))dx
= ∫ (x(x-1)!/(2x-1))dx
= [((x^2 - x + 1)/(4x^2 - 2x)) * (x-1)!] + C
The integral does not have a closed-form expression, so it is not possible to determine whether the integral converges or diverges by traditional methods. Thus, we cannot conclude that the series converges or diverges by the integral test.
c) Since we couldn't determine convergence or divergence using the integral test, we can't select this option.
d) The nth term test: The nth term test states that if the limit as n approaches infinity of the nth term of the series does not equal zero, then the series diverges.
Let's find the limit of the nth term:
lim (n->∞) (n!/(2n-1))
= ∞/(∞ - 1)
= ∞
The limit is not zero, so the nth term test implies that the series diverges.
So, among the options provided, the correct choice is d) The series diverges by the nth term test.