Find the limit when $n \rightarrow \infty$ of the series: $$\frac n{n^2}+\frac n{n^2+1^2}+ \frac n{n^2+2^2}+\cdots+\frac 1{n^2+(n+1)^2}$$
I am required to do this using limit of a sum definition of the integral. However, to me it seems that the final fraction should have been $\frac n{n^2+(n+1)^2}$ instead of $\frac 1{n^2+(n+1)^2}$ since that would allow the answer to be given by: $$\lim_{n \to \infty}\sum_{k=0}^{n+1}\frac{n}{n^2+k^2}=\lim_{n \to \infty}\frac{1}{n}\sum_{k=0}^{n+1}\frac{1}{1+(\frac k{n})^2} =\int_0^1\frac{1}{1+x^2}dx=\frac \pi{4}$$
I was a little unsure of whether the question was actually wrong or whether I overlooked something. Then, I had a look at the solution in the book and, turns out, even the solutions don't match. The book says the answer is $\frac 1{4}\log 2$. Clearly I'm missing something here. If someone could tell me where I'm going wrong that'd be great.