The link you attached confirms what I applied in my attempt at the problem, however, I am still confused on why the Alternating Series Test failed and how to correct this.
You appear to be forgetting to take the absolute value of each term. You are using negative [imath]b_n[/imath]:The link you attached confirms what I applied in my attempt at the problem, however, I am still confused on why the Alternating Series Test failed and how to correct this.
(Note: I attempted to prove the summation is conditionally convergent using the following theorem.)Alternating_series_test: If [imath]a_1\ge a_2\ge a_3\cdots a_n\ge \cdots\ge 0[/imath]
such that [imath]\left(a_n\right)\to 0[/imath] then the series [imath]\displaystyle\sum\limits_{n = 1}^\infty {{{( - 1)}^n}{a_n}} [/imath] converges.
In common parlance: if the sequence part of an alternating series is a decreasing to zero the series converges.
Can you explain why you think that does not apply here?
No, the alternating series test does apply! You are just applying it incorrectly because of the difference you point out.I noticed that the the series is not (-1)^n , but (-1)^(n-1). Is this discrepancy why the Alternating Series Test does not apply?