05/31/2011, 03:02 AM
the series converges for large values of x, [e, 6] was the value where x begins to converge.
I do not have a proof of its convergence, that's why I posted it here; I only managed to make it this far (I'm not good with limits).
\( L = \lim_{n\to\infty} |x (\frac{1}{n+1} - \frac{1}{(n+1)^2\ln(n)})| \)
the series should converge as long as \( 0 \le L \lt 1 \), but I'm pretty sure that limit works out to L = 0, for at least positive real x. I must've made a mistake somewhere therefore.
Perhaps this is a quasi-power series.
\( f(x) = e^x \ln(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!} \psi_0(n+1) \) even though \( \psi_0(n+1) \neq f^{(n)}(0) \)
Right now I'm not sure what else to suggest. This leaves me a little in wonderment.
I do not have a proof of its convergence, that's why I posted it here; I only managed to make it this far (I'm not good with limits).
\( L = \lim_{n\to\infty} |x (\frac{1}{n+1} - \frac{1}{(n+1)^2\ln(n)})| \)
the series should converge as long as \( 0 \le L \lt 1 \), but I'm pretty sure that limit works out to L = 0, for at least positive real x. I must've made a mistake somewhere therefore.
Perhaps this is a quasi-power series.
\( f(x) = e^x \ln(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!} \psi_0(n+1) \) even though \( \psi_0(n+1) \neq f^{(n)}(0) \)
Right now I'm not sure what else to suggest. This leaves me a little in wonderment.
