I just came across a question on an old take-home exam,
Prove, using induction, that $\sum_{k = 1}^{n} \frac{1}{k^2} > \frac{3}{2} - \frac{1}{n} + \frac{1}{2n^2} $
Then I remembered something that the professor said about his method of coming up with these problems by looking at graphs and integrals (and possibly partitions/lower sums ??).
How can we generate similar proofs by induction (where, say, the denominators have degree at most $= 3$)?
Can you demonstrate such a method with an example(s)?
I think we're looking for inequalities here...
*Note: I don't need help proving this inequality. *