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

Question:

Determine whether the series is conditionally convergent, absolutely divergent, or divergent. State what test you are using (i.e., Integral Test, Divergent Test...) for each step.{eq}\displaystyle \sum \limits_{n=1}^{\infty} \frac { (-1)^n }{2n + sin (n) } {/eq} (Give a neat and detailed step)

Alternating Series Test:

To check for convergence and divergence of the series {eq}\sum\limits_{n = 1}^\infty {{l_n}} {/eq}, the series can be represented as {eq}{l_n} = {( - 1)^{n + 1}}{k_{n\,\,\,\,\,\,\,}}{\text{or }}{l_n} = {( - 1)^n}{k_{n\,\,\,\,\,}} {/eq}, where {eq}{k_n} \geqslant 0,\forall n,\, {/eq}.

The following conditions must be satisfied for the series to converge.

(1){eq}\mathop {\lim }\limits_{n \to \infty } {k_n} = 0 {/eq}

(2){eq}\left\{ {{k_n}} \right\} {/eq} is a decreasing sequence.

Answer and Explanation:

Given that: {eq}\displaystyle \sum\limits_{n = 1}^\infty {\frac{{{{( - 1)}^n}}}{{2n + sin(n)}}} {/eq}

{eq}\displaystyle \eqalign{ &...

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