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 March 6th, 2018, 11:10 AM #1 Senior Member   Joined: Dec 2015 From: somewhere Posts: 550 Thanks: 83 Show inequality true Show that $\displaystyle (1+ \frac{1}{n} )^n < (1+ \frac{1}{n+1})^{n+1} \;$ for $\displaystyle n$ natural
 March 6th, 2018, 12:48 PM #2 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,950 Thanks: 1141 Math Focus: Elementary mathematics and beyond Make an attempt and post your work. Are you expected to use any particular method, such as induction?
 March 8th, 2018, 12:21 PM #3 Senior Member   Joined: Dec 2015 From: somewhere Posts: 550 Thanks: 83 Since $\displaystyle (1+\frac{1}{n})^n=f(n)1$ use Bernoulli's inequality $\displaystyle (1+r)^x>1+xr$ with conditions $\displaystyle [1+(-(n+1)^{-2}))]^{1+n}>1+(-(n+1)^{-2}\Rightarrow (1+\frac{1}{n})^n < (1+\frac{1}{n+1})^{n+1}$

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