126 lines
1.6 KiB
TeX
126 lines
1.6 KiB
TeX
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\section{More about $e$}
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\problem{}
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Show that
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$$
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\lim_{n\to\infty}{
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\bigg(
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1 + \frac{1}{n+1}
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\bigg)^n
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= e
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}
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$$
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\vfill
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\problem{}
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Show that
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$$
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\lim_{n\to\infty}{
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\bigg(
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1 + \frac{1}{n}
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\bigg)^{n+1}
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= e
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}
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$$
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\vfill
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\problem{}<inverse_e>
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Show that
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$$
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\lim_{n\to\infty}{
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\bigg(
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1 - \frac{1}{n}
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\bigg)^n
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= \frac{1}{e}
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}
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$$
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\begin{solution}
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$
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\lim_{n\to\infty}{(1 - \frac{1}{n})^n} =
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\lim_{n\to\infty}{(\frac{n-1}{n})^{(-1)(-n)}}
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$ \par
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$
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= \lim_{n\to\infty}{(\frac{n}{n- 1 })^{-n}}
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= \lim_{n\to\infty}{(1 + \frac{1}{n-1})^{-n}}
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$ \par
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$
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= \lim_{n\to\infty}{{(1 + \frac{1}{n-1})^{(n-1)(\frac{n}{n-1})}}^{-1}}
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$ \par
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$
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= \frac{1}{e}
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$
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\end{solution}
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\vfill
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\problem{}
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Show that
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$$
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\lim_{n\to\infty}{
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\bigg(
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1 + \frac{x}{n}
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\bigg)^n
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= e^x
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}
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$$
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Note that \ref{inverse_e} is a special case of this problem.
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\vfill
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\pagebreak
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\theorem{}
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The following important formula is proven in most calculus courses.
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$$e^x = \sum_{n=0}^{\infty}{\frac{x^n}{n!}} = 2 + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$$
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\vfill
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\problem{}
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What are the first six digits of $e$?
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\begin{solution}
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$e = 2.718\ 281\ 828$
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\end{solution}
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\vfill
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\definition{}
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If $f$ is a function, we say that $L$ is a limit of $f$ at $\infty$ if for every $\epsilon > 0$, we can find an $M \in \mathbb{R}$ so that $|f(x) - L| < \epsilon$ for $x > M$. \par
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If this is true, we say that $L = \lim_{x\to\infty}{(f(x))}$.
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\vfill
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\problem{}
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Prove the following: \par
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Hint: If $x > 0$, then $\lfloor x \rfloor \leq x \leq \lceil x \rceil$
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$$ \lim_{x\to\infty}{\bigg(1 + \frac{1}{x}\bigg)^x} = e$$
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\vfill
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\pagebreak
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