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How many terms of the series do we need to add in order to find the sum to the indicated...

Question:

How many terms of the series do we need to add in order to find the sum to the indicated accuracy?

{eq}\sum_{n = 1}^\infty \frac{(-1)^{n - 1}}{n^2} {/eq}, error {eq}\leq 0.006. {/eq}

Alternating Series Test:

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

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

Then for convergent of the series following condition satiesfied;

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

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

Answer and Explanation: 1

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

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

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