Why is this break not executing for the summation of pi

pie = 0;
for k = 0:100
    eqn = sqrt(12)*((-3)^(-k)/((2*k)+1));
    PrevSum = pie;
    pie = pie + eqn;
    if PrevSum ~= 0 & (PrevSum/pie) < (10^(-8))
        k
        return
    end
end
pie
PrevSum/pie
>>
pie =

   3.141592653589794


ans =

     1

[Alec Bischoff]

PrevSum/pie should eventually get below 10^-8, but my code doesn't seem to.

Answer 1

[Stephan]

You never reach the value of your ratio of PrevSum/pie. I inserted 2 lines, the first saves the ratio in every single run of your loop, the second plots the ratio over the variable k. This helps you to understand whats going wrong. You will have to think about another criteria to stop the loop, when the maximum deviation is reached:

pie = 0;
for k = 0:100
    eqn = sqrt(12)*((-3)^(-k)/((2*k)+1));
    PrevSum = pie;
    pie = pie + eqn;
    error_rate(k+1) = PrevSum/pie;
    if PrevSum ~= 0 && (PrevSum/pie) < (10^(-8))
        k
        return
    end
end
pie
PrevSum/pie
plot(0:100,error_rate)

Answer 2

[Alan Stevens]

PrevSum/pie should tend to 1. abs(PrevSum - pie) should get smaller than 10^-8.

Try

pie = 0;
k = 0;
err = 1;
while err>10^-9
    PrevSum = pie;
    eqn = sqrt(12)*((-3)^(-k)/((2*k)+1));
    pie = PrevSum + eqn;
    err =abs(PrevSum/pie - 1); 
    k = k+1;
end
format long
pie
PrevSum/pie
abs(PrevSum - pie)