Approximate $(0.99)^{300}$ without calculator.
This question is in my textbook but i don't know how to approximate without calculator. How can i evaluate without calculator? Thanks in advance.
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$\begingroup$Remark that we can write:$$ (0.99)^{300} = (1-0.01)^{300} = \left( 1 + \frac{-3}{300} \right)^{300} $$
Now, recalling that we have:$$ e^x = \lim \limits_{n \rightarrow \infty} \left( 1 + \frac{x}{n} \right)^{n} $$
We conclude that we can approximate $(0.99)^{300}$ as $e^{-3}$.
$\endgroup$ 5 $\begingroup$$$300 \ln (1-1/100) \approx 300 (-1/100-1/20000) \approx -3$$
$$e^{-3} = (3-(3-e))^{-3} \approx \frac{1}{27} \left(1+(3-e)\right)=\frac{4-e}{27}=0.0475...$$
$$0.99^{300}=0.0490...$$
As for "without calculator", using $e=2.718...$ is enough. If you remember how to divide by hand.
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