Sunday, 6 November 2016

real analysis - Prove 2ncdotn!(n+1)n by induction.



An induction I'm struggling with.




Prove 2nn!(n+1)n by induction.





An idea was to show that 2nn!1+n2 since 1+n2(n+1)n using Bernoulli. However the inequality is just wrong so that approach doesn't work. I had the intuition that 2nn! but I don't think that yields anything for this problem.



I would really like to get a hint or two. Of course you can post your answer, this is obviously what this platform is for, but I won't read them until I solved the problem myself. It's an induction, can't be that difficult right?


Answer



Hint: The induction step goes as follows:
2n+1(n+1)!=2nn!2(n+1)(n+1)n2(n+1)=2(n+1)n+1
Thus you are left to prove that 2(n+1)n+1(n+2)n+1, which is pretty easy.


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