i simply cant get the the limit comparison test to work on this following function,
Summation of (from 2 to infinity)[ln(k)^(5/2)/k^(15/8)]
Please someone help!
$\displaystyle \forall x > 0,\,\ln (x) < x \Rightarrow \quad \ln \left( {k^{\frac{5}{4}} } \right) < k^{\frac{5}{4}} \Rightarrow \quad \frac{5}{4}\ln (k) < k^{\frac{5}{4}} $
$\displaystyle \frac{{\left[ {\ln (k)} \right]^{\frac{5}{2}} }}{{k^{\frac{{15}}{8}} }} < \frac{{\left( {\frac{4}{5}} \right)^{\frac{2}{5}} k^{\frac{1}{2}} }}{{k^{\frac{{15}}{8}} }} = \frac{{\left( {\frac{4}{5}} \right)^{\frac{2}{5}} }}{{k^{\frac{{11}}{8}} }}
$
i am still frustrated... i didnt see how you got from step 3 to step 4 and how did u get $\displaystyle
\frac{{\left[ {\ln (k)} \right]^{\frac{5}{2}} }}{{k^{\frac{{15}}{8}} }} < \frac{{\left( {\frac{4}{5}} \right)^{\frac{2}{5}} k^{\frac{1}{2}} }}{{k^{\frac{{15}}{8}} }} = \frac{{\left( {\frac{4}{5}} \right)^{\frac{2}{5}} }}{{k^{\frac{{11}}{8}} }}
$
O.K.
Start over. Consider this:$\displaystyle \left[ {\ln (k)} \right]^{\frac{5}{2}} = \left[ {10\ln \left( {k^{\frac{1}{{10}}} } \right)} \right]^{\frac{5}{2}} < \left( {10} \right)^{\frac{5}{2}} \left( {k^{\frac{1}{{10}}} } \right)^{\frac{5}{2}} = \left( {10} \right)^{\frac{5}{2}} \left( {k^{\frac{1}{4}} } \right)$