Question

Does this series converge or diverge? (alternating series test)

Answer 1

\[\sum_{n=1}^{\infty}\frac{ (-1)^{n+1} }{ 7n^8 +2 }\]

Answer 2

Well, you don't even need an alternating series test for this series, well, unless you asked to use precisely this test.

Answer 3

Alternating Series Test:

Suppose you have a series in a form.
\(\color{#000000}{ \displaystyle \sum_{ n=k }^{ \infty } (-1)^nA_n}\) or \(\color{#000000}{ \displaystyle \sum_{ n=k }^{ \infty } (-1)^{n-1}A_n}\).
(which either one of these is same in terms of convergence)

If \(A_n\) is (an always positive and) a monotonically decreasing sequence,
meaning that: \(\color{#000000}{ \displaystyle A_{n+1} <A_n}\) for all \(n\), THEN the series converges.

Answer 4

For example:

\(\color{#000000}{ \displaystyle \sum_{ n=3 }^{ \infty }(-1)^n\frac{1}{n}\quad \Longrightarrow~~ {\rm converges.}}\)
Because the series alternates and the non-alternating part (1/n) in this case, always decreases, the series will converge.

Answer 5

You can re-write the series in your case as,
\(\color{#000000}{ \displaystyle \sum_{ n=0 }^{ \infty }(-1)^{n+1}\frac{1}{7n^8+2}}\)

Answer 6

The series is alternating, and the non-alternating part is always decreasing (because the denominator gets larger and larger causing the term to get smaller and smaller).