Base Case

August 21st, 2003

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Proving an inequality using mathematical induction?

Hey guys, the question is here: http://img84.imageshack.us/img84/7789/inductionif7.jpg

I've done the base case and the induction hypothesis... but I'm not really sure how to go about the induction conclusion. Thanks for any help you can give!

So you have that:

1 + 1/2^2 + ... + 1/n^2 <= 2 - 1/n (*) and you want to prove that implies 1 + 1/2^2 + ... + 1/n^2 + 1/(n+1)^2<= 2 - 1/(n+1) (**) To convert (*) into (**) lets first work on making both left sides equal, so lets take (*) and add 1/(n+1)^2 to both sides of the inequality, to get: 1 + 1/2^2 + ... + 1/n^2 + 1/(n+1)^2<= 2 - 1/n + 1/(n+1)^2 (***) Now turn to the right side of (***): 2 - 1/n + 1/(n+1)^2 = 2 + (n-(n+1)^2) / (n*(n+1)^2) = = 2 + (n - n*(n+1) -(n+1)) / (n*(n+1)^2) = = 2 + (-n*(n+1) -1) / (n*(n+1)^2) = = 2 - 1/(n+1) - 1 / (n*(n+1)^2) <= 2 - 1/(n+1) Which completes your proof by induction...

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