Question

∑1∞(-1)^n/n^1/3
determines the series whether converges conditionally, converges absolutely or diverges, justify your answer.

Answer 1

Does the series \[\sum_{n=1}^{\infty}|1/n^{\frac{1}{3}}|\]
converge? If it does it converges absolutely, otherwise, it converges conditionally at best. (Hint: this is a p-series and convergence depends on the power of n)

Answer 2

Use the alternating series test.

For a series of the form \(\sum (-1)^na_n\), the series converges if
\[(1)~~~~a_n\text{ is a decreasing sequence}\\
(2)~~~~a_n\to0\]
Additionally, the series absolutely converges if \(\sum |(-1)^na_n|\) also converges.

Answer 3

tq