Calc6.7

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Explain why the integral test is or is not applicable to \sum _{{n=1}}^{{\infty }}{\frac  {|\sin n|+1}{\ln(n+1)}}.

This series contains terms which are all positive since the numerator and denominator are always positive. This function is also always continuous, though it has many sharp turns. However, this function is never decreasing. The sine function causes this series to continually increase for a short while and then decrease for a short while, increase for a short while and then decrease for a short while, and so on. Thus, we can not use the integral test on this series. However, the direct comparison test shows us that this series diverges.


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