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+ [tex]\frac{x}{n}[/tex] ) ^n < e^x
at the first, i take log on both sides,.. but i couldn't go further.
can someboday help me?
thx
To prove this inequality, you can use mathematical induction or the binomial theorem. Alternatively, you can also use the Taylor series expansion of e^x and compare it to the given expression.
Yes, the proof involves breaking down the given expression into smaller components and then using algebraic manipulations and mathematical properties to compare it to the Taylor series expansion of e^x. It is important to carefully justify each step in the proof.
No, the inequality is true for all n ∈ ℕ (natural numbers). However, as n increases, the inequality becomes more accurate.
Yes, there are multiple ways to prove this inequality. Some other methods include using the Mean Value Theorem or the Intermediate Value Theorem.
This inequality has various applications in mathematics and physics, such as in the study of exponential growth and decay. It can also be used to approximate values in engineering and economics problems.