MATLAB: Limit to infinity of Left Riemann Sum

infinite series function left riemann sum

Hi there,
Lets say I have created a function Sn=LeftRiemannSum(f,left,right,N) ,that computes the left riemann sum over the interval left to right with N subdomains.i.e.:
Sn=sum(f(xi)*h) for all subdomains i=0 to N-1. f is my function and xi=left+i*h , so the input arguments left=x0 and right=xN.
Let f be my anonymous function (ex f=@(x)(x.*log(1+x)) .
I also estimated the Sn for varying N, from N=10 to 100000.
Now, I simply want to compute the value of the series Sn when N -> infinity. Inside the function I have a for loop [ for i=0:(N-1)] so I will have endless loop ..
Can I pass the function somehow to the 'limit' command? Any clues?
Thanx!
PS: The main part of the code of my function LeftRiemann Sum is the following:
if true
  for i=0:(N-1)
    x=x0+i.*h;
    y=f(x);
    A=y.*h;
    S=S+A
  end
   Sn=S
end

Best Answer

If I am not mistaken, you try to determine the limit of the left Riemann-sum for the value of the definite integral. Of course you can't take infinite members. If you want to use this approach, I recommend you to use a large number for N and also estimate the right Riemann-sum. If the two sums are approximately equal, then there is hope that this is the approximate value of the integral.
A comment: it can easily be vectorized:
N = 1000;
a = 1;
b = 2;
h = (b-a)/N;
x = x0+(0:N)*h;
fx = f(x);
sum(fx*h);
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