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We know that $$\sum a_k \text{ converges} \iff \text{the partial sums } s_n \text{converge} \iff \text{the partial sums } s_n \text{are Cauchy}$$

Writing out what this last statement means

$$\forall \varepsilon \gt 0, \exists N, \text{such that } \forall m \ge n \gt N, \left \lvert \sum_{k=n}^{m} a_k \right \rvert \lt \varepsilon$$

Let $\displaystyle a_k = \frac{1}{k ^{3.5}}$ and let $\displaystyle \varepsilon = 10^{−4}$.

Find a value of N that satisfies the Cauchy condition written out above.

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    Could compare the sums to (simpler to evaluate) integrals.2012-03-08
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    @max What do you know about $\sum k^{-2}$? What do you now about $k^{-3.5}$ vs $k^{-2}$ when $k>1$?. What does this tell you about the ultimate value of your sum?2012-03-08
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    You can make us of Euler-Maclaurin summation to come with really tight error bounds. http://en.wikipedia.org/wiki/Euler-Maclaurin_formula2012-05-08

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