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# AQA C2 pg 155 watch

1. Flicking through an old book I came across this problem

By comparing the area under the graph of 1/(x^2) between x - r-1 and r with thw area of an appropriate rectangle over the same interval

show that (1/(r^2) < integral between (r-1) and r of 1/(x^2) DONE THAT
Deduce that sum between r=1 and N of 1/(r^2) < integral between (1) and N of 1/(x^2) DONE THAT

But can't do the following

Hence show that sum between r=1 and N of 1/(r^2) < 2

2. (Original post by maggiehodgson)
Flicking through an old book I came across this problem

By comparing the area under the graph of 1/(x^2) between x - r-1 and r with thw area of an appropriate rectangle over the same interval

show that (1/(r^2) < integral between (r-1) and r of 1/(x^2) DONE THAT
Deduce that sum between r=1 and N of 1/(r^2) < integral between (1) and N of 1/(x^2) DONE THAT

But can't do the following

Hence show that sum between r=1 and N of 1/(r^2) < 2

Show that the integral is always less than 2 regardless of what the value of N is.
3. That's the bit I can't do.

I get, for the integral, 1/(r(r-1)) and then what?
4. (Original post by maggiehodgson)
That's the bit I can't do.

I get, for the integral, 1/(r(r-1)) and then what?
You should get the integral as -1/x and then sub in the limits of N and 1 and then it should be clear.
5. if the integral is 2 and the rectangle sum is less than the integral then the rectangle sum is less than 2
6. BUT I've got to show that

the integral is < 2 regardless of the value of N. don't think I can say "let the integral = 2" as a starting point.
7. A sketch of the curve and thinking about the summation as rectangles and the integral as the total area under the curve gives the following inequality .
Now this inequality involves but you want the inequality for and it's just a one step manipulation which gives the result they want.
8. (Original post by B_9710)
A sketch of the curve and thinking about the summation as rectangles and the integral as the total area under the curve gives the following inequality .
Now this inequality involves but you want the inequality for and it's just a one step manipulation which gives the result they want.
Thanks

I think that has sorted me out.

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