using induction, prove: 1/1^2 + 1/2^2 + 1/3^2 + ... + 1/n^2 < 2 - 1/k?

Question

I can do the base case and formulate the induction hypothesis (of course), but I'm not sure exactly how to prove this one. Inequalities give me a hard time. Please prove this as an example for me if you can. Thanks

Answer 1

I assume you mean < 2 - 1/n?

For the induction step, assume that

1/1^2 + 1/2^2 + 1/3^2 + ... + 1/n^2 < 2 - 1/n

for some n. we need to show

1/1^2 + 1/2^2 + ... + 1/n^2 + 1/(n+1)^2 < 2 - 1/(n+1).

Subtract the first inequality from the second, and you find that you must show

1/(n+1)^2 < 1/n - 1/(n+1).

1/n - 1/(n+1) = (n+1)/(n+1)(1/n) - 1/(n+1)(n/n)
= [(n+1)-n]/n(n+1) = 1/n(n+1) > 1/(n+1)^2

so the above inequality does in fact hold.