Determine whether the series is absolutely convergent, conditionally convergent, or divergent. Show all steps

Question

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. Show all steps

Answer 1

Use the so-called "p test"...i.e. when does 1/n^p converge.

If you don't know the answer to that question, you should go back to your textbook and read about this test.

Answer 2

∑_{(n=1 to ∞)} l (-1)ⁿ(1 / (n⁹+1) l = ∑_{(n = 1 to ∞)} [ 1 / (n⁹ + 1) ] < ∑_{(n = 1 to ∞)} [1/n⁹]

The series ∑_{(n=1 to ∞)} [ 1 / n⁹ ] is a convergent p-series with p = 9.

So, by the Direct Comparison Test, ∑_{(n = 1 to ∞)} [ 1 / (n⁹ + 1) ] converges.

Therefore, the series ∑_{(n = 1 to ∞)} [ (-1)ⁿ ( 1 / (n⁹+1)) ] converges absolutely.