while loop not taking decimal or fractional answers and unrecognized variable

Hi there

I am trying to write a while loop to determine convergence of a Taylor and/or Laurent series. I am not being assigned to do this, I just want to do it for myself as it would be very useful. I've found codes online for "normal" series, but I can't find one that works for a form of power series. On top of this though, I am having issues with the "normal" series code. I have started with this code I found online.

ER=100; 
s=0;
n=1;
while (ER >= 0.01)
   n_sum = (1/(n^2));
   s = s + n_sum; %compute the running sum       
   ER =(abs(((pi^2)/6)-s)/((pi^2)/6))*100; %Calculate error
   n = n + 1; %counter 
end
disp(s)
disp(n-1)
disp(ER)

This is very useful as it calculates the error, number of times it took to converge, and the point it converged to. I want to make this more general and input 1.6448 rather than (pi^2)/6. I made the change as follows:

ER=100; 
s=0;
n=1;
while (ER >= 0.01)
   n_sum = (1/(n^2));
   s = s + n_sum; %compute the running sum       
   ER =(abs(((1.6448)-s)/((1.6448))*100; %Calculate error
   n = n + 1; %counter 
end
disp(s)
disp(n-1)
disp(ER)

This, however, gives me an "invalid expression, when calling a function or indexing a variable, use parentheses. Otherwise, check for mismatched dlimiters." If I enter it as a fraction, it also does not work. I am struggling to understand why. I attempted changing the error as follows as I read somewhere that there could be an issue with symbolic language? I don't know what that is as I'm fairly new, but I tried, and it didn't work.

ER=100; 
s=0;
n=1;
while (double(subs(ER)) >= 0.01)
   n_sum = (1/(n^2));
   s = s + n_sum; %compute the running sum       
   ER =(abs(((1.6448)-s)/((1.6448))*100; %Calculate error
   n = n + 1; %counter 
end
disp(s)
disp(n-1)
disp(ER)

This also significantly slowed down the code if I put (pi)^2/6 back in.

The main goal of this is also causing errors. If I try to input a Laurent series as follows:

ER=100; 
s=0;
n=1;
while (double(subs(ER)) >= 0.01)
   n_sum = (x/(n*(x-n)));
   s = s + n_sum; %compute the running sum       
   ER =(abs(((1.6448)-s)/((1.6448))*100; %Calculate error
   n = n + 1; %counter 
end
disp(s)
disp(n-1)
disp(ER)

I am also getting an error: "Unrecognized variable 'x'." I can't fix this as x is not defined. It's the variable in the power series. Sorry for all of the code, I would have uploaded photos if I could. Any help is appreciated!

Answer 1

[Walter Roberson]

   ER =(abs(((1.6448)-s)/((1.6448))*100; %Calculate error
       1   234      3  2 34      32

The digit is the number of unmatched open-brackets "after" the character above it.

I am also getting an error: "Unrecognized variable 'x'." I can't fix this as x is not defined. It's the variable in the power series.

Are you trying to build a formula ? If so then you can use the Symbolic Toolbox -- but if you do, then you cannot calculate an absolute error -- the absolute error is going to depend upon the input x.

I am trying to write a while loop to determine convergence of a Taylor and/or Laurent series.

You cannot use that approach for unknown inputs.

For known inputs, you can calculate whether a new term is going to be insignificant relative to the current sum, but that approach cannot be used to demonstrate convergence. Consider for example the series sum(1/n, n, 1, 10^16-1) which is roughly 37.418577 . The next term would be 10^(-16) which would be less than eps(37.418577) and therefore cannot add anything to the double precision number, and all remaining terms are even smaller . Using a while loop to test the value of the term would imply that sum(1/n,n,1,inf) is convergent.. but it is not. sum(1/n,n,1,x) converges to log(x)+eulernumber and log(inf) is infinity . So even though if you calculate in double precision going forward from n=1, the terms after 10^16 cannot add anything to the sum, mathematically those remaining terms to n=infinity add up to infinity.

You need a different approach if you want to establish convergence.

[Dominic Blanco]

Could you suggest to me how I could go about that? I have been struggling to find one. My main goal is to have convergence of Taylor and Laurent series.

[Walter Roberson]

syms x n m
S = symsum(x/(n*(x-n)),n,1,m)

S = 

R = rand() * 5

R = 2.3960

Sx = subs(S, x, R)

Sx = 

limit(Sx, m, inf)

ans = 

subs(S, x, 7)

ans = 

limit(ans,m,inf)

ans = 

NaN

So no convergence if x is a positive integer. If it is not, then you need to prove that psi(m+1)-psi(m-x+1) converges . And to use your approach, you would have to do the proof of convergence using a while loop . For all permitted non-integer x.

You need a different approach if you are wanting to prove it for all permitted (non-integer) x.

[Dominic Blanco]

Got it. And this makes sense because (x-n) will be 0 if x is an integer, and that will cause a singularity. Thank you for your help! I will try to modify this to make it work for x non-integers.