Interesting Limit

Calculus Level 4

$\left \lfloor \lim_{n\to \infty } \left ( \frac{1000}{n+1}+\frac{1000}{n+2}+\cdots +\frac{1000}{n+n} \right) \right \rfloor = \, ?$

The answer is 693.

1 solution

Dwaipayan Shikari
Feb 3, 2021

$\lim_{n\rightarrow{∞}} 1000(\dfrac{1}{n+1}+\dfrac{1}{n+2}+\cdots+\dfrac{1}{n+n})= \lim_{n\rightarrow{∞}}\dfrac{1000}{n} \sum_{r=1}^n \dfrac{1}{1+\dfrac{r}{n}}$ Applying Riemann sums It is $1000\int_0^1 \dfrac{1}{1+x} dx = [1000 \log(1+x)]_0^1$ $= 1000\log(2)= 693.147\cdots$ Answer is $⌊1000\log(2)⌋=693$

Interesting Limit

The answer is 693.

1 solution

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