In fact, the two terms you input do not converge. So, if you enter the expression in that form, the program will meet two infinities and can't add them up.
In addition, if a limit
$ \underset{n \rightarrow \infty}{\lim} a_n - b_n $ exists (converges), it doesn't mean series
$\underset{n\rightarrow \infty}{\lim}a_n$ and
$\underset{n\rightarrow \infty}{\lim}b_n$ both exist (converge).
The two
$\infty$ in you expression may be different precisely. The expression you input is equal to
$ \underset{n\rightarrow \infty}{\lim}\sum_{k=0}^{n} \frac{ln(k+2)}{2(k+2)} - \underset{m\rightarrow \infty}{\lim}\sum_{k=0}^{m}\frac{ln(k+1)}{2(k+1)} $
where two series both don't converge actually. So the sum of them don't converge.
But if you sum each term together like the reply before, you will get a series that converge and is easy to calculate the result.
$ \underset{n\rightarrow \infty}{\lim}(\sum_{k=0}^{n} \frac{ln(k+2)}{2(k+2)} - \sum_{k=0}^{n}\frac{ln(k+1)}{2(k+1)}) = \underset{n\rightarrow \infty}{\lim}\sum_{k=0}^{n}( \frac{ln(k+2)}{2(k+2)} - \frac{ln(k+1)}{2(k+1)}) $
Here the two
$\infty$ are the same and we can add each term of two series up. The series above converge and is equal to 0.
Addition: the two infinite series are different actually. The first one doesn't converge but the second one converges to a finite value.
