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Series Convergence: Two-sided Estimator

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A simple technique to estimate values of the finite sums of reciprocals to natural numbers raised to positive power and to define if corresponding infinite series converge.

Deduce two-sided estimator for the sums of the positive powers (p) of reciprocals to natural numbers
to prove convergence of corresponding series for p more than 1, and divergence of the series for [EQUATION].

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A two-sided estimator is used to prove divergence of a summation.
The solution is detailed and well presented.

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Solution.

To deduce the two-sided estimator for
= ,
we plot the graph of the function for x  0:

Total square of all rectangles with vertexes below the curve is equal to and is obviously less than the value of the ...

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