The Importance of Limits in the Development of Mathematics

[sadhu]

how can you say that L$$^{1}$$=1

unless you show it maths wise

 

[titaniumx3]

how can you say that L$$^{1}$$=1

unless you show it maths wise

I'm not entirely sure how you'd show it rigourously and in all honesty I'm not sure if it makes any sense at all with regards to L_1 but,

1 = sqrt(1) = sqrt(sqrt(1)) = sqrt(sqrt(sqrt(1))) = ...

That said, the above is using only the representation of L_1 in my previous post, not in the a_1 = sqrt(n), a_(k+1) = sqrt(n+a_K) representation (which I claim equals L_2, not L_1).

Maybe it is just poorly written notation, but the way L_1 is definined in my last post doesn't make any sense at all because:

$$L_{1} = \sqrt{0+\sqrt{0+\sqrt{0+...}}} = \sqrt{\sqrt{\sqrt{...}}}$$

implies there is no end term, so how on Earth do you know it is equal to 0? Personally I think it is best to define it simply as $a_{1}=\sqrt{n},a_{k+1}=\sqrt{n+a_{k}}$ and leave it there.

 

[titaniumx3]

May I ask, when someone discusses "limiting processes", what exactly does that mean? Does it simply refer to things like the sum of an infinite series, calculation of differentials/integrals? What about taking the limit of a function or sequence, is that also a limiting process?

BTW, I'd like to thank whoever recommended the book, The Origins Of Cauchy's Rigorous Calculus; it's an interesting read, easy to comprehend and invaluable to the topic at hand.