How do you apply the ratio test to determine if #sum (1*3*5* * * (2n1))/(1*4*7* * * (3n2))# from #n=[1,oo)# is convergent to divergent?
the series is convergent.
We can apply d'Alembert's ratio test:
Suppose that;
Then
if L < 1 then the series converges absolutely; if L > 1 then the series is divergent; if L = 1 or the limit fails to exist the test is inconclusive.
So our series is;
So our test limit is:
And so the series is convergent.
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To apply the ratio test to determine the convergence or divergence of the series ( \sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n1)}{1 \cdot 4 \cdot 7 \cdot \ldots \cdot (3n2)} ), follow these steps:

Compute the ratio of consecutive terms: [ r = \lim_{n \to \infty} \left \frac{a_{n+1}}{a_n} \right ] where ( a_n ) represents the ( n )th term of the series.

Substitute the expressions for ( a_n ) and ( a_{n+1} ) into the ratio: [ r = \lim_{n \to \infty} \left \frac{(2(n+1)1)}{(3(n+1)2)} \cdot \frac{(3n2)}{(2n1)} \right ]

Simplify the expression and compute the limit.

If the limit ( r ) is less than 1, the series converges. If ( r ) is greater than 1 or the limit does not exist, the series diverges.

Based on the result obtained in step 4, conclude whether the series ( \sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n1)}{1 \cdot 4 \cdot 7 \cdot \ldots \cdot (3n2)} ) is convergent or divergent.
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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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