The positive integer just greater than (1+0.0001)^10000 is:

Question

The positive integer just greater than $(1+0.0001)^{10000}$ is:

Accepted Answer

Let $x=10000$. $\left(1+\dfrac1x\right)^x=\displaystyle\sum_{k=0}^x\binom xk\dfrac1{x^k}=1+1+\dfrac1{2!}\left(1-\dfrac1x\right)+\dfrac1{3!}\left(1-\dfrac1x\right)\left(1-\dfrac2x\right)+\cdots$

$$<1+\frac1{1!}+\frac1{2!}+\frac1{3!}+\cdots=e<3$$

Since the value exceeds 2 (comparing the first few terms) and is less than $e<3$, the value lies in $(2,3)$, so the smallest integer greater than it is 3.

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