Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm struggling around my homework. I hope someone will point me the right direction for solving following examples:

  • Prove that $n^{n+1} > (n + 1)^n$ for $n > 2$;

  • Prove that $(1 + x)^n \ge 1 + nx$; $x \in\Bbb R$; $n \in\Bbb N$;

  • Prove that $(2n)! < 2^{2n}(n!)^2$; $n \ge 1$;

  • Prove that $2^n > n$:

Thank you a lot.

share|cite|improve this question

closed as off-topic by 6005, Mathmo123, Grigory M, Adam Hughes, quid Jan 7 '15 at 22:39

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – 6005, Mathmo123, Grigory M, Adam Hughes, quid
If this question can be reworded to fit the rules in the help center, please edit the question.

Show us what you did and where you are stuck. – Chris Gerig Sep 28 '12 at 21:49
So, do you not know what induction is? What do you know about induction? – Graphth Sep 28 '12 at 21:54
up vote 3 down vote accepted

For the first note that the inequality is equivalent to $n>(1+\frac1n)^n$

For the second note that $(1+nx)(1+x)=1+(n+1)x + n x^2\ge 1+(n+1)x$ and use this in an induction proof at least for the case $x\ge -1$. Think about it: Is the claim true at all if we allow very negative $x$? What about $x=-3$ and $n=5$?

For the third: Replace each odd factor $k$ in $(2n)!$ by $k+1$ and extract $2n$ factors of $2$ from the $2n$ (now) even factors.

The third is a very trivial induction.

share|cite|improve this answer
Did you mean fourth in that last line? – Brian M. Scott Sep 28 '12 at 21:53
While I realize the word has a (soft) mathematical meaning, it's kind of mean to call others' problems trivial. – Snowball Sep 28 '12 at 22:07
It is not mean if it is indeed trivial, which it is. – copper.hat Sep 28 '12 at 23:38

Not the answer you're looking for? Browse other questions tagged or ask your own question.