How do you test the alternating series #Sigma ((-1)^(n+1)2^n)/(n!)# from n is #[0,oo)# for convergence?

Question

Cameron Alexander · Accepted Answer

Converges absolutely

Use the ratio test. This test works for alternating series. If there is a series #sum_(n=0)^infty a_n# then consider #lim_(n rarr infty)abs(a_(n+1)/a_n)=L#. If #L<1# then the series absolutely converges. If #L>1# then the series diverges. If #L=1# then the test is inconclusive. For our series, we have: #lim_(n rarr infty)abs(((-1)^((n+1)+1)2^(n+1))/((n+1)!)div((-1)^(n+1)2^n)/(n!))# #lim_(n rarr infty)abs(((-1)^(n+2)2^(n+1)n!)/((-1)^(n+1)2^n(n+1)!)# #lim_(n rarr infty)abs((-2)/(n+1))# #0# #0<1# so the series converges absolutely.

Ezekiel Alexander · Answer

undefined 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.

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

3. **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.