In these days I'm studying about real analysis and Improper Riemann integral. And my problem is mainly based on series and sequences on above sections. If we mainly consider about a function like f(x) = $\frac{1}{n^p} $ in both sections like,
- In Integration, $\rhd$ $\int_0^\infty \! f(x) \, \mathrm{d}x$
- In Real Analysis, $\rhd$ $\sum_{n=1}^\infty\!f(x)$
So in here my lecturer got the convergence of above sequences different in each particular sections in this way,
$$\begin{array}{c|c|c|} & \text{In Real Analysis} & \text{In Improper Riemann Integral} \\ \hline \text{When p > 1} & converge & diverge \\ \hline \text{When 1 $\geq$ p > 0} & diverge & converge \\ \hline \end{array}$$
Can anyone please explain why the convergence is happening totally different in these sections ?
And my other question is above peculiar condition is not valid for tests in each particular sections like limit comparison test, Direct comparison test, absolute convergence.
Although f(x) changing was different we consider same conditions for above tests in real analysis and Improper Riemann integral.
Thanks in Advance