My Math Forum Equation with Landau's "o"

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 November 14th, 2017, 03:09 AM #1 Newbie   Joined: Oct 2017 From: Italy Posts: 15 Thanks: 0 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?
 November 14th, 2017, 02:33 PM #2 Global Moderator   Joined: May 2007 Posts: 6,665 Thanks: 651 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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