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A) Use the alternating series test to determine whether the series is convergent or divergent....

Question:

A) Use the alternating series test to determine whether the series is convergent or divergent.

{eq}\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n3^{n}} {/eq}

B) If the series is convergent, use the alternating series estimation theorem to determine how many terms we need to add in order to find the sum with an error less than 0.0001.

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}.

Then for convergence of the series, following conditions must be satisfied;

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

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

Answer and Explanation:

Given that: {eq}\displaystyle \sum\limits_{n = 1}^\infty {\frac{{{{( - 1)}^n}}}{{n{3^n}}}} {/eq}

{eq}\displaystyle \eqalign{ & \sum\limits_{n =...

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