How do you test the series #Sigma (100^n(n!)^3)/((3n)!)# from #n=[1,oo)# by the ratio test?
The series:
is divergent.
The ratio test states that given the series:
and the limit:
Let's calculate the ratio for our series:
So:
and the series is divergent.
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To test the series Σ (100^n(n!)^3)/((3n)!) using the ratio test:

Compute the ratio of consecutive terms: R = a_{n+1} / a_n where a_n represents the nth term of the series.

Evaluate the limit of R as n approaches infinity.

Determine convergence based on the limit of R:
 If the limit is less than 1, the series converges.
 If the limit is greater than 1 or does not exist, the series diverges.
 If the limit equals 1, the test is inconclusive.
Now, apply the ratio test to the given series:
a_n = (100^n(n!)^3) / ((3n)!)
a_{n+1} = (100^(n+1)((n+1)!)^3) / ((3(n+1))!)
R = (100^(n+1)((n+1)!)^3) / ((3(n+1))!) / (100^n(n!)^3) / ((3n)!)
R = (100^(n+1)((n+1)!)^3)(3n)! / (100^n(n!)^3)(3(n+1))!)
R = (100(n+1))^3 / (3(n+1))(3(n+1)1)(3(n+1)2)
R = (100(n+1))^3 / (3(n+1))(3n+2)(3n+1)(3n)
Now, take the limit of R as n approaches infinity:
lim (n→∞) R = lim (n→∞) [(100(n+1))^3 / (3(n+1))(3n+2)(3n+1)(3n)]
As n approaches infinity, the terms involving n in the denominator dominate, leading to a limit of 0.
Since the limit is less than 1, the series Σ (100^n(n!)^3)/((3n)!) converges by the ratio test.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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