How do you test the alternating series #Sigma ((1)^(n+1)2^n)/(n!)# from n is #[0,oo)# for convergence?
Converges absolutely
Use the ratio test. This test works for alternating series.
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To test the convergence of the alternating series ( \sum_{n=0}^{\infty} \frac{(1)^{n+1}2^n}{n!} ), you can use the Alternating Series Test. This test states that if the terms of an alternating series decrease in absolute value and approach zero as ( n ) approaches infinity, then the series converges.

Check for Absolute Convergence: First, consider the absolute value of the terms of the series: [ \left \frac{(1)^{n+1}2^n}{n!} \right = \frac{2^n}{n!} ]

Apply the Ratio Test: Next, apply the Ratio Test to determine if the series ( \sum_{n=0}^{\infty} \frac{2^n}{n!} ) converges absolutely. The Ratio Test states that if ( \lim_{n \to \infty} \left \frac{a_{n+1}}{a_n} \right = L < 1 ), then the series converges absolutely.
For ( a_n = \frac{2^n}{n!} ), compute: [ \lim_{n \to \infty} \left \frac{a_{n+1}}{a_n} \right = \lim_{n \to \infty} \left \frac{\frac{2^{n+1}}{(n+1)!}}{\frac{2^n}{n!}} \right = \lim_{n \to \infty} \left \frac{2}{n+1} \right = 0 ] Since the limit is less than 1, the series ( \sum_{n=0}^{\infty} \frac{2^n}{n!} ) converges absolutely.

Verify Terms Decrease: Finally, observe that the terms ( \frac{2^n}{n!} ) are positive and decreasing as ( n ) increases.
Since the series ( \sum_{n=0}^{\infty} \frac{2^n}{n!} ) converges absolutely, and the terms alternate in sign in the original series, by the Alternating Series Test, the alternating series ( \sum_{n=0}^{\infty} \frac{(1)^{n+1}2^n}{n!} ) converges.
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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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