How do you use the ratio test to test the convergence of the series #∑(2k)!/k^(2k) # from n=1 to infinity?
Take the limit as
Then, multiplying gives us
. . .
Simplifying, we get
Then we substitute back to obtain
Thus
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To use the ratio test for testing the convergence of the series ∑(2k)!/k^(2k) from n=1 to infinity, we compute the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th term to the nth term. The series converges if this limit is less than 1, and diverges if it is greater than 1 or inconclusive if it equals 1.
Applying the ratio test to the given series:
We have the general term a_n = (2n)! / n^(2n)
So, the (n+1)th term is a_(n+1) = (2(n+1))! / (n+1)^(2(n+1))
Taking the ratio of the (n+1)th term to the nth term:
R = [(2(n+1))! / (n+1)^(2(n+1))] / [(2n)! / n^(2n)]
R simplifies to: R = [(2n+2)(2n+1)] / [(n+1)^2]
Taking the limit of R as n approaches infinity:
lim (n→∞) |R| = lim (n→∞) |[(2n+2)(2n+1)] / [(n+1)^2]|
= lim (n→∞) |(4n^2 + 6n + 2) / (n^2 + 2n + 1)|
= 4
Since the limit is 4, which is greater than 1, the ratio test suggests that the series diverges. Therefore, the series ∑(2k)!/k^(2k) from n=1 to infinity diverges.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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