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

Question

Ezra Adams · Accepted Answer

#sum_(n=1)^oo(-1)^(n+1)(1+1/n)# diverges.

The alternating series test basically says this: If a series is alternating, then as long as the ABSOLUTE VALUE of each term is less than the ABSOLUTE VALUE of the previous term, the series will converge. BUT, we must first check that #lim_(n->oo)a_n = 0# (the nth term test). #lim_(n->oo)(-1)^(n+1)(1+1/n) = +-1 != 0# Not only do the terms of this series NOT approach #0# as #n -> oo#, but also they approach the series #sum_(n=1)^oo(-1)^n# which is known to be divergent. Therefore, the series diverges. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~``` NOTE: If you forget to check using the n-th term test before using the Alt. Series Test, you will usually get the wrong answer. For example, we could have proven that each term of this series is closer to #0# than the last term, so the series meets the criteria for the Alt. Series Test. We would then falsely assume convergence even though the series approaches a divergent sum.

Cameron Alexander · Answer

undefined To test the alternating series $ \sum_{n=1}^{\infty} (-1)^{n+1} \left(1 + \frac{1}{n}\right) $ for convergence, you can use the Alternating Series Test. This test states that if the series alternates in sign and the absolute value of its terms decreases monotonically to zero, then the series converges.

1. Check the alternating sign: The series alternates in sign because it has the term $ (-1)^{n+1} $, which changes sign as $ n $ increases.

2. Check the monotonic decrease: Consider the absolute value of the terms of the series, which is $ \left|1 + \frac{1}{n}\right| = 1 + \frac{1}{n} $. As $ n $ increases, $ \frac{1}{n} $ decreases, so $ 1 + \frac{1}{n} $ also decreases monotonically to $ 1 $.

Since the series alternates in sign and the absolute value of its terms decreases monotonically to zero, the Alternating Series Test guarantees convergence. Therefore, the series $ \sum_{n=1}^{\infty} (-1)^{n+1} \left(1 + \frac{1}{n}\right) $ converges.