10 questions linked to/from Prove that $e^x\ge x+1$ for all real $x$
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### Sum of exponential function inequality

For any positive a and n, it seems this inequality holds $$\sum\limits_{t=n+1}^\infty e^{-at} \leq \frac{1}{a}e^{-an}$$ How can I prove this inequality and does this holds for negative a ?
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### Simplest or nicest proof that $1+x \le e^x$

The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or ...
### For the function $y=\ln(x)/x$: Show that maximum value of y occurs when $x = e\ldots$
For the function $y=\ln(x)/x$: Show that maximum value of $y$ occurs when $x = e$. Using this information, show that $x^e <e^x$ for all positive values of $x$. Two positive integers, $a$ and $b$,...