Discrete and continuous problems

[haiha]

Hi all,

There are some dificult problems with discrete argument n that will be very easy if I can change it to continuous argument x. But I do not know what is the condition for that.
For example: to calculate the sum of a1+a2 +a3+...an. when n goes to infinity, can I make it as S=integral of a(x) from xo to infinity ?

Thanks

 

[mathman]

For example: to calculate the sum of a1+a2 +a3+...an. when n goes to infinity, can I make it as S=integral of a(x) from xo to infinity ?

It depends very much on how you define a(x) for x not an integer.

 

[haiha]

Could you please explain more about that.
I just think the sume of An is similar to the way we calculate the integral in calculus. I mean Sum(An) = a1*d+a2*d+...an*d where d equals 1. When we calculate to rather large integer n, then d = 1 is small enough to have adequate accuracy.
May I think that this can happen only when f(x) must have positive (or negartive) derivative all over the domain, and f(x) must also have no critical points. ?

 

[mathman]

Since a(x) has to be defined for x not integer, the integral will depend on precisely how it is defined. For example, linear interpolation between the integer values will give you a function which can be integrated to give the same result as the summation.