Determine the convergence of the series Σ(n=1 to ∞) (n!)²/(2n)! using the ratio test.
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Step-by-Step Solution
Step 1: State the Ratio Test
The ratio test is a powerful tool to determine the convergence or divergence of an infinite series. It involves calculating the limit of the absolute ratio of consecutive terms. If this limit L is less than 1, the series converges. If L is greater than 1 or infinite, the series diverges. If L equals 1, the test is inconclusive.
Step 2: Identify an and an+1
First, we need to clearly identify the n -th term, an, of the given series. Then, we derive the (n+1) -th term, an+1, by replacing n with (n+1) in the expression for an. Remember that (n+1)!=(n+1)⋅n! and (2n+2)!=(2n+2)(2n+1)(2n)!.
Step 3: Form the Ratio anan+1
Now, we form the ratio anan+1. This involves dividing the (n+1) -th term by the n -th term, which is equivalent to multiplying the (n+1) -th term by the reciprocal of the n -th term. This step sets up the expression for simplification.
Step 4: Simplify the Ratio
We simplify the ratio by expanding the factorials. We know that (n+1)!=(n+1)n! and (2n+2)!=(2n+2)(2n+1)(2n)!. After substituting these expansions, many terms cancel out, leading to a much simpler algebraic expression.
Step 5: Calculate the Limit
To find the limit as n approaches infinity, we can divide both the numerator and the denominator by the highest power of n, which is n. As n→∞, terms like n1 and n2 approach zero. This simplifies the limit calculation significantly.
Step 6: Conclusion
Since the limit L is 41, which is less than 1, according to the ratio test, the series converges. This means that the sum of the terms of the series approaches a finite value.