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November 14th, 2017, 03:09 AM   #1
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Equation with Landau's "o"

Find $a,b,c \in \mathbb{R}$ such that

$$e^{\sqrt{n+2}-\sqrt{n}} -1 + cos(n^{-4})=a+\frac{b}{\sqrt{n}} + \frac{c}{n}=o(\frac{1}{n})$$


I put $e^{\sqrt{n+2}-\sqrt{n}} \longrightarrow 1$

and $-1 + cos(n^{-4})=-\frac{1}{2 \sqrt{n}} + o(\frac{1}{n})$, so maybe $a=1$ and $b=-\frac{1}{2}$ but it's wrong!

(The solutions must be $a=1$, $b=\frac{1}{2}$ and $c=\frac{13}{24}$)

Any hint?
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November 14th, 2017, 02:33 PM   #2
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You need to expand the exponential to get the terms in $\displaystyle 1/\sqrt{n}$ powers.
$\displaystyle cos(n^{-4})=1-\frac{1}{2n^4}+...$
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