Question

Prove that $1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+.....+\frac{1}{n^2}<2-\frac{1}{2}$ for all $n>2$, n $ϵ$ N.

Answer 1

Let P(n) : $1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+.....+\frac{1}{n^2}<2-\frac{1}{2}$ for all $n\le 2$

For n = 2

$1+\frac{1}{4}<2-\frac{1}{4}$

$=\frac{5}{4}<\frac{7}{4}$

$\Rightarrow$ P(n) is true for n = 2.

Let P(n) is true for n = k.

$1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+.....+\frac{1}{k^2}<2-\frac{1}{(k+1{)}^2}<2-\frac{1}{(k+1)}$

Now,

$1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+.....+\frac{1}{k^2}+\frac{1}{(k+1{)}^2}<2-\frac{1}{k}+\frac{1}{(k+1{)}^2}$ [Using (1)]

$<2-\frac{k^2+2k+1-k}{k(k+1{)}^2}$

$<2-\frac{k^2+k+1}{k(k+1{)}^2}$

$<2-\frac{k^2+k}{k(k+1{)}^2}$

$<2-\frac{k(k+1)}{k(k+1{)}^2}$

$<2-\frac{1}{k+1}$

$\Rightarrow$ P(n) is true for n = k + 1

$\Rightarrow$ P(n) is true for all $nϵN$ by PMI