Question

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

Answer 1

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.