Code:
\[ C_{n}-C_{n+1}=\sum_{k=n^{2}}^{(n+1)^{2}-1}\frac{1}{k^{a}}-\sum_{k=(n+1)^{2}}^{(n+2)^{2}-1}\frac{1}{k^{a}}=\sum_{k=n^{2}}^{(n+1)^{2}-1}\frac{1}{k^{a}}-\sum_{k'=n^{2}}^{(n+1)^{2}+1}\frac{1}{(k'+2n+1)^{a}}=\sum_{k=n^{2}}^{(n+1)^{2}-1}\left(\frac{1}{k^{a}}-\frac{1}{(k+2n+1)^{a}}\right)-\frac{1}{((n+2)^{2}-2)^{a}}-\frac{1}{((n+2)^{2}-1)^{a}}=\sum_{k=0}^{2n}\left(\frac{1}{(n^{2}+k^{)}^{a}}-\frac{1}{(n+1)^{2}k^{a}}\right)-\frac{1}{((n+2)^{2}-2)^{a}}-\frac{1}{((n+2)^{2}-1)^{a}}> (2n+1)\left (\frac{1}{(n^{2}+2n)^{a}}-\frac{1}{((n+1)^{2}+2n)^{a}}\right)-\frac{1}{((n+2)^{2}-2)^{a}}-\frac{1}{((n+2)^{2}-1)^{a}} \]