How can I prove that x−x22<ln(1+x) for any x>0
I think it's somehow related to Taylor expansion of natural logarithm, when:
ln(1+x)=x−x22+x33−⋯
Can you please show me how? Thanks.
Answer
Hint:
Prove that ln(1+x)−x+x22 is strictly increasing for x>0.
edit: to see why this isn't a complete proof, consider x2−1 for x>0. It's strictly increasing; does that show that x2>1? I hope not, because it's not true!
No comments:
Post a Comment