Problem was
1) Show that π4>√2−√2
2) Show that ∑∞n=21n2>35
For part 1) we can give a geometric proof of sorts.
1) Show that π4>√2−√2
2) Show that ∑∞n=21n2>35
For part 1) we can give a geometric proof of sorts.
A circle of radius 1. The length of arc AB is π4. The length of the line segment AB is 2sinπ8=√2−√2. Since arc length > line segment length, we are done.
For part 2)
We use the following inequality
n2<(n−13)(n+23)
This gives us
1n2>1n−13−1n+23
Setting n=2,3,… and adding we get a telescoping series on the right and the only term that remains is 12−13=35.
This implies part 1 because of the classic result that ∑∞n=11n2=π26.
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