Unit
- 181
- 0
I read this in the appendix of Hardy's pure mathematics textbook and an explanation was not given:
\lim_{x \to 1} \sum_1^\infty \left( x^{n-1}-x^n \right) = \lim_{x \to 1} \left( 1 - x + (x - x^2) + \ldots \right) = \lim_{x \to 1} 1 = 1
\sum_1^\infty \lim_{x \to 1} \left( x^{n-1}-x^n \right) = \sum_1^\infty \left( 1 - 1 \right) = 0 + 0 + 0 + \ldots = 0
Are both answers correct? I thought that associativity only applied to a finite series of operations, which is why I think the first equation is wrong.
\lim_{x \to 1} \sum_1^\infty \left( x^{n-1}-x^n \right) = \lim_{x \to 1} \left( 1 - x + (x - x^2) + \ldots \right) = \lim_{x \to 1} 1 = 1
\sum_1^\infty \lim_{x \to 1} \left( x^{n-1}-x^n \right) = \sum_1^\infty \left( 1 - 1 \right) = 0 + 0 + 0 + \ldots = 0
Are both answers correct? I thought that associativity only applied to a finite series of operations, which is why I think the first equation is wrong.