Could you show me $$\mathop {\lim }\limits_{x \to 0} \left( {1 + \sum\limits_{n = 1}^\infty {{{\left( { - 1} \right)}^n}{{\left( {\frac{{\sin nx}}{{nx}}} \right)}^2}} } \right) = \frac{1}{2}.\tag{1}$$
These day, I want to write a article about the sums of divergent series. In Hardy's book Divergent Series, he show me a method proposed by Riemann with $$\mathop {\lim }\limits_{x \to 0} \left( {1 + \sum\limits_{n = 1}^\infty {{{\left( { - 1} \right)}^n}{{\left( {\frac{{\sin nx}}{{nx}}} \right)}^2}} } \right) ,$$ from which we can obtain $$1-1+1-1+\cdots=\frac12.$$ I hope someone tell me how to prove (1),Thanks!