Induction on inequalities: $\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\ldots+\frac1{n^2}<2$

Question

I am trying to solve this inequality by induction. I just started to learn induction this week and all the inequalities we had been solved were like an equation less than another equation (e.g. $n! \geq 2^n$) So I am confused how to prove an inequality that is less than a particular value like the following problem? Thanks in advance.

$$\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\ldots+\frac1{n^2}<2$$

Answer 1

A useful fact about proofs by induction is that sometimes it’s easier to prove a stronger result. That happens to be the case here. Let $$s_n=\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\ldots+\frac1{n^2}\;.$$

Then

$$\begin{align*} 2-s_1&=1\ge 1\;,\\ 2-s_2&=\frac34\ge\frac12\;,\\ 2-s_3&=\frac34-\frac19=\frac{23}{36}\ge\frac13\;,\text{ and}\\ 2-s_4&=\frac{23}{36}-\frac1{16}=\frac{83}{144}\ge\frac14\;. \end{align*}$$

Those numbers $1,\frac34,\frac{23}{36}$, and $\frac{83}{144}$ are the ‘amounts of space’ left between $s_1,s_2,s_3$, and $s_4$ on the one hand, and the bound of $2$ on the other. They don’t seem to be shrinking very fast: $\frac{83}{144}$ is still $0.5763\overline{8}$. It looks as if they might be shrinking slower than the simple sequence $1,\frac12,\frac13,\frac14,\ldots\;$; certainly they are so far, and we have

$$\begin{align*} s_1&=1\le 2-1\;,\\ s_2&\le 2-\frac12\;,\\ s_3&\le 2-\frac13\;,\text{ and}\\ s_4&\le 2-\frac14\;. \end{align*}$$

This suggests that just maybe $s_n\le 2-\dfrac1n$ for all $n\ge 1$. If true, that would certainly imply that $s_n<2$.

HINT: Try to prove by induction that $s_n\le 2-\dfrac1n$ for $n\ge 1$. Note that $$s_{n+1}=s_n+\frac1{(n+1)^2}<s_n+\frac1{n(n+1)}$$ (why?): this will be useful for the induction step.

Answer 2

Hint: Instead of trying to prove that

$$\dfrac{1}{1^2}+ \dfrac{1}{2^2}+\cdots+\dfrac{1}{n^2} < 2,$$

try instead to prove the stronger inequality

$$\dfrac{1}{1^2}+ \dfrac{1}{2^2}+\cdots+\dfrac{1}{n^2} \le 2-\dfrac{1}{n}.$$

This is easily done by induction.

Hint 2: The inequality $\dfrac{1}{(n+1)^2} < \dfrac{1}{n(n+1)}$ will be helpful.

Answer 3

$$\frac{1}{k^2}＜\frac{1}{k(k-1)}$$

$$\frac{1}{1^2}+\frac{1}{2^2}+\cdots+\frac{1}{n^2}=1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}＜1+\frac{1}{2(2-1)}+\cdots+\frac{1}{n(n-1)}=2-\frac{1}{n}＜2$$