This

sum((-1)^n/sqrt(n)*cos(ln(n)),{n,1,4000})

WolframAlphaInput

seems to strongly hint that will contribute a factor of about -0.643 to the result.

This

sum(1/sqrt(n)*cos(ln(n)),{n,1,4000})

WolframAlphaInput

does not strongly hint what that sum will be, but runs out of time for larger n.

Wait to see if the graphs appear after it says it has run out of time doing the calculations.

This

lim 1/n as n->infinity

is the notation that Wolfram|Alpha usually understands when trying to enter a limit

but I have not been able to coax Wolfram|Alpha into finding your limit.

Part of the problem may be that Sum may be expecting your n to be an integer but that Lim is not expecting m to be an integer