- #1
missavvy
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Hey guys... I just did this question... and I have the solution but I do not understand it still. If someone could just explain what happened.
1 + 1/4 + ... 1/n2 < 2 - 1/n
For n>=2
So the base case:
n=2:
1/4 < 2- 1/2, 1/4 < 3/2 True
Suppose this holds for n:
1+ 1/4 +... 1/n2 + 1/(n+1)2 < 2 - 1/n + 1/(n+1)2
Here is where I don't understand what happens..
Since 1/n - 1/(n+1) = 1/n(n+1) > 1/(n+1)2 , we have 2 - 1/n + 1/(n+1)2 < 2 - 1/(n+1).
Then the inequality for n+1 follows.
So I'm confused with why we use 1/n - 1/(n+1).. where did it come from ?
Thanks!
Homework Statement
1 + 1/4 + ... 1/n2 < 2 - 1/n
For n>=2
Homework Equations
The Attempt at a Solution
So the base case:
n=2:
1/4 < 2- 1/2, 1/4 < 3/2 True
Suppose this holds for n:
1+ 1/4 +... 1/n2 + 1/(n+1)2 < 2 - 1/n + 1/(n+1)2
Here is where I don't understand what happens..
Since 1/n - 1/(n+1) = 1/n(n+1) > 1/(n+1)2 , we have 2 - 1/n + 1/(n+1)2 < 2 - 1/(n+1).
Then the inequality for n+1 follows.
So I'm confused with why we use 1/n - 1/(n+1).. where did it come from ?
Thanks!