In Wolfram Alpha entry 1 the following is claimed to be true:
$$\lim_{n\to \infty } \left(\sum _{k=1}^n \log (2 k)-\sum _{k=1}^{n+\frac{1}{2}} \log (2 k-1)\right)=\frac{1}{2} \log \left(\frac{\pi }{2}\right)$$
What is the meaning of adding a fraction like $\frac{1}{2}$ in the upper summation limit in the second sum, inside the limit?
Is it a bug a or a feature?
Compare to Wolfram Alpha entry 2 where the fraction $\frac{1}{2}$ in the upper summation limit of the second sum has been left out:
$$\lim_{n\to \infty } \left(\sum _{k=1}^n \log (2 k)-\sum _{k=1}^{n} \log (2 k-1)\right)=\infty$$
I get the same results in Mathematica 8.0.1:
Limit[Sum[Log[2*k], {k, 1, n}] - Sum[Log[2*k - 1], {k, 1, n + 1/2}],
n -> Infinity]
Limit[Sum[Log[2*k], {k, 1, n}] - Sum[Log[2*k - 1], {k, 1, n}],
n -> Infinity]
and as well as:
Limit[Sum[Log[2*k], {k, 1, n - 1/4}] - Sum[Log[2*k - 1], {k, 1, n + 1/4}], n -> Infinity]
Limit[Sum[Log[2*k], {k, 1, n}] - Sum[Log[2*k - 1], {k, 1, n + 1/4}], n -> Infinity]
The only support I was able to find in the documentation for Sum
is:
"The limits of summation need not be numbers. They can be Infinity or symbolic expressions."