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November 14th, 2017, 02:09 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, 01:33 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,511 Thanks: 585 
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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