Determine whether the series is absolutely convergent. 1-(1...

Determine whether the series is absolutely convergent.
$$\begin{aligned}
1-\frac{1 \cdot 3}{3 !}+\frac{1 \cdot 3 \cdot 5}{5 !}-\frac{1 \cdot 3 \cdot 5 \cdot 7}{7 !}+\\
+(-1)^{n-1} \frac{1 \cdot 3 \cdot 5 \cdot \quad \cdot(2 n-1)}{(2 n-1) !} +
\end{aligned}$$

Key Concepts

Absolute Convergence

Absolute convergence is a property of an infinite series where the series formed by taking the absolute values of its terms converges. When this happens, it implies the original series converges, and often the analysis of absolute convergence is easier because one can ignore any alternating signs. This concept is central when assessing convergence since a series converges absolutely if and only if the corresponding series of absolute values converges.

Ratio Test

The ratio test is a method used to determine the convergence of a series by examining the limit of the absolute value of the ratio of consecutive terms. If this limit is less than one, the series converges absolutely, while if it is greater than one, the series diverges. This test is particularly useful in series involving factorials or products of sequences, as it simplifies the handling of terms that grow rapidly.

Factorial Growth

Factorial growth refers to the rapid increase in the value of factorials as their argument becomes large. Factorials grow faster than exponential functions and many polynomial functions, which often leads to convergence in series where the denominator contains a factorial term. Comparing the product in the numerator with the factorial in the denominator can show how the terms decay as n increases.

Alternating Series

An alternating series is one in which successive terms alternate in sign. While the alternating nature can sometimes be used to establish conditional convergence through tests like the alternating series test, assessing absolute convergence requires ignoring the alternating signs and directly testing the convergence of the absolute values of the terms.

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