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Convergence or Divergence of the Series

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For the following either prove it or give a counterexample for 1 and 2

1. If sum(a_n) with a_n>0 is convergent. then is sum([a_n] ^2) always convergent.
2. If sum(a_n) with a_n>0 is convergent. then is sum([a_n a_(n+1)] ^0.5] always convergent.

3. If sum(a_n) with a_n>0 is convergent. and b_n= (a_1 + ...+ a_n)/n for n in the naturals then show that sum(b_n) is always divergent
4. Establish the convergence or divergence of the series whose nth term is ((n^2)(n+1))^-0.5

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Solution Summary

Establish the convergence or divergence of the series.

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