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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4$\begingroup$ Are you claiming that the textbook in which this appears as an exercise provides no information about how to do it? $\endgroup$– José Carlos SantosNov 6, 2018 at 17:42
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1$\begingroup$ @Delta-u I'm pretty sure that approximation ($(1+x)^r\sim1+rx$) fails due to how large $300$ is. $\endgroup$– Rushabh MehtaNov 6, 2018 at 17:45
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2$\begingroup$ I approximate it as $1$. $\endgroup$– The CountNov 6, 2018 at 17:53
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3$\begingroup$ @TheCount So, if I roll a 100 sided dice 300 times, there's a 100% chance I won't get any 1s? $\endgroup$– Alexander Gruber ♦Nov 6, 2018 at 17:58
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1$\begingroup$ @TheCount My point was that there is really only a <5% chance, but you are correct on the second point-- moderators have no concept of humor $\endgroup$– Alexander Gruber ♦Nov 7, 2018 at 6:43
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2 Answers
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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 $e^{-3}$.
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$\begingroup$ This is nice, I didn't notice the significance of $300$ $\endgroup$– Yuriy SNov 6, 2018 at 18:07
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$\begingroup$ Basically, if $n$ is large, then the approximation is good. 300 is "pretty big", so the approximation is "pretty good". $\endgroup$– SamboNov 6, 2018 at 18:29
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$\begingroup$ Thanks a lot. It really saved my time. This method is very helpful. $\endgroup$ Nov 7, 2018 at 6:49
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$\begingroup$ I assume this is what the textbook was going for; it seems like a way to introduce students to the idea of $e$. $\endgroup$– SamboNov 7, 2018 at 14:08
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$\begingroup$ @ishakbayrak If you think this answers your question, you can click the check mark to mark this answer as "accepted". $\endgroup$– SamboNov 9, 2018 at 19:14
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$$300 \ln (1-1/100) \approx 300 (-1/100-1/20000) \approx -3$$
$$e^{-3} = (3-(3-e))^{-3} \approx \frac{1}{27} \left(1+(3-e)\right)=\frac{4-e}{27}=0.0475...$$
$$0.99^{300}=0.0490...$$
As for "without calculator", using $e=2.718...$ is enough. If you remember how to divide by hand.
