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Determine if the following series are absolutely convergent, conditionally convergent, or...

Question:

Determine if the following series are absolutely convergent, conditionally convergent, or divergent.

(a) {eq}\sum \frac{(-1)^n\sqrt n}{n^3+4} {/eq}.

(b) {eq}\sum (-1)^n \arctan (\frac{1}{n}) {/eq}.

Alternating Series Test:

Let to check for convergence and divergence of the series {eq}\sum\limits_{n = 1}^\infty {{h_n}} {/eq} .

Then series can be represnted as {eq}{h_n} = {( - 1)^{n + 1}}{q_{n\,\,\,\,\,\,\,}}{\text{or }}{h_n} = {( - 1)^n}{q_{n\,\,\,\,\,}} {/eq}, where {eq}{q_n} \geqslant 0,\forall n,\, {/eq}.

Then for convergent of the series following condition satiesfied;

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

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

Absolute Convergence:

{eq}\displaystyle \sum {{a_k}{\text{ is absolute convergence if }}} \sum {\left| {{a_k}} \right|{\text{ is convergent}}{\text{.}}} {/eq}

Conditionally Convergent:

{eq}\displaystyle \sum {{a_k}{\text{ is conditionally convergent if }}} \sum {{a_k}{\text{ is convergent and }}} \sum {\left| {{a_k}} \right|{\text{ is divergent}}{\text{.}}} {/eq}

Answer and Explanation:

{eq}\displaystyle \eqalign{ & a) \cr & \sum {\frac{{{{( - 1)}^n}\sqrt n }}{{{n^3} + 4}}} \cr & {\text{Let,}} \cr & {a_n} = \frac{{{{( -...

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