How do you test for convergence for #(-1)^(n-1) /(2n + 1)# for n=1 to infinity?

1 Answer
Bill K.
May 27, 2016

The series #sum_{n=1}^{infty} ((-1)^(n-1))/(2n+1)# converges by the Alternating Series Test .

Explanation:

Let #a_{n}=1/(2n+1)#. Then the given series is in the form #sum_{n=1}^{infty}(-1)^{n-1}a_{n}#. Since #a_{n}\geq 0# for all natural numbers #n# and is a sequence that monotonically decreases to zero, the Alternating Series Test implies that #sum_{n=1}^{infty} ((-1)^(n-1))/(2n+1)# converges.

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