42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from j = 0 to ∞)2 ⋅ 4ʲ / (2j + 1)!

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Identify the given series: \( \sum_{j=0}^{\infty} \frac{2 \cdot 4^{j}}{(2j + 1)!} \). We want to determine if this infinite series converges or diverges.

Consider using the Ratio Test, which is often effective for series involving factorials and exponential terms. The Ratio Test states that for \( a_j \), if \( L = \lim_{j \to \infty} \left| \frac{a_{j+1}}{a_j} \right| \), then the series converges if \( L < 1 \), diverges if \( L > 1 \), and is inconclusive if \( L = 1 \).

Write the general term \( a_j = \frac{2 \cdot 4^{j}}{(2j + 1)!} \) and the next term \( a_{j+1} = \frac{2 \cdot 4^{j+1}}{(2(j+1) + 1)!} = \frac{2 \cdot 4^{j+1}}{(2j + 3)!} \).

Form the ratio \( \left| \frac{a_{j+1}}{a_j} \right| = \frac{2 \cdot 4^{j+1}}{(2j + 3)!} \times \frac{(2j + 1)!}{2 \cdot 4^{j}} = \frac{4 \cdot (2j + 1)!}{(2j + 3)!} \). Simplify the factorial expression to express the ratio in terms of \( j \).

Evaluate the limit \( L = \lim_{j \to \infty} \left| \frac{a_{j+1}}{a_j} \right| \) using the simplified expression. Based on the value of \( L \), conclude whether the series converges or diverges according to the Ratio Test.

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Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Determining whether a series converges means checking if the sum approaches a finite limit as the number of terms grows indefinitely. Understanding convergence is essential to analyze the behavior of series like the one given.

Ratio Test

The Ratio Test is a common method to determine the convergence of series with factorials or exponential terms. It involves taking the limit of the absolute value of the ratio of consecutive terms. If this limit is less than one, the series converges absolutely.

Factorials and Growth Rates

Factorials (n!) grow faster than exponential functions as n increases. Recognizing how factorials dominate growth helps in applying convergence tests effectively, especially when terms include factorials in the denominator, often leading to convergence.

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.∑ (from j = 0 to ∞)2 ⋅ 4ʲ / (2j + 1)!