Determine the convergence of the series Σ($n=1$ to ∞) (n!)²/(2n)! using the ratio test.

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 $a_n$ and $a_{n+1}$

First, we need to clearly identify the $n$ -th term, $a_n$, of the given series. Then, we derive the $(n+1)$ -th term, $a_{n+1}$, by replacing $n$ with $(n+1)$ in the expression for $a_n$. Remember that $(n+1)! = (n+1) \cdot n!$ and $(2n+2)! = (2n+2)(2n+1)(2n)!$.

Step 3: Form the Ratio $\frac{a_{n+1}}{a_n}$

Now, we form the ratio $\frac{a_{n+1}}{a_n}$. 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 $n \to \infty$, terms like $\frac{1}{n}$ and $\frac{2}{n}$ approach zero. This simplifies the limit calculation significantly.

Step 6: Conclusion

Since the limit $L$ is $\frac{1}{4}$, 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.

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