Which of the following is the value of the limit \lim_{n\to∞} Σ_{k=0}^{∞}{n-1} \frac{1}{n+k}?

Question

Which of the following is the value of the limit \( \lim\limits_{n\to\infty} \) \( \sum\limits_{k=0}^{n-1}{} \) \( \frac{1}{n+k} \) ?

(Hint: Use Riemann sums.)

a.ln3

b.Does not exist.

c.ln2

d.∞

e.2/3

Answer 1

Siehe https://www.wolframalpha.com/input/?i=sum+1%2F%28n%2Bk%29%2C+k%3D0+to+n-1%2C+n+to+infinity

Answer 2

You can use Riemann sums as stated in the hint.
\(\begin{aligned} \lim_{n \to\infty} \sum_{k=0}^{n-1} \frac{1}{n+k} &= \lim_{n \to\infty} \sum_{k=0}^{n-1} \left(\frac{n}{n+k}\cdot \frac{2-1}{n}\right) = \lim_{n \to\infty} \sum_{k=0}^{n-1} \left(\left(\frac{1}{1+\frac{k}{n}}\right)\cdot \frac{2-1}{n}\right)\\&= \int_{1}^{2} \ln\left(x\right) \mathrm{d}x = \ln\left(2\right) \end{aligned}\)
whereby we considered a partition of the interval \([1, 2]\) into \(n\) equally spaced subintervals.