How many terms is needed to estimate the sum of Sigma_{n = 1}^{infinity} (-1)^n / n^2 + 4 so that...

Question:

How many terms is needed to estimate the sum of {eq}\sum\limits_{n = 1}^{\infty} \frac{(-1)^n}{n^2 + 4} {/eq} so that the error is {eq}< 0.001? {/eq}

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^2} + 4}}} {/eq}

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

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