The inequality (1 + x/n)^n < e^x can be proven using the binomial theorem and power series expansion for e^x. By applying logarithms to both sides, the left-hand side can be expanded, revealing that for x > 0, the inequality holds clearly. For x < 0, a more nuanced approach is required, involving the manipulation of terms in pairs to establish the inequality. This method provides a definitive proof of the stated inequality.
PREREQUISITESMathematicians, students studying calculus, and anyone interested in proving inequalities involving exponential functions.
Expand the lhs by the binomial theorem and take the power series for e^x. The solution becomes obvious for x>0. For x<0, it is a little trickier - work with terms 2 at a time.wowolala said:the quesion is below, show that
show that ( 1+ \frac{x}{n} ) ^n < e^x
at the first, i take log on both sides,.. but i couldn't go further.
can someboday help me?
thx