Okay so I managed to launch it (FINALLY) But (it would have been to easy) I have now these errors

Table::iterb: Iterator {i, 2, n} does not have appropriate bounds.

Table::iterb: Iterator {i, 1 + n, -1 + 2 n} does not have appropriate bounds.

Table::iterb: Iterator {i, 1 + n, -1 + 2 n} does not have appropriate bounds.

General::stop: Further output of Table::iterb will be suppressed during this calculation.

Part::pkspec1: The expression i cannot be used as a part specification.

Part::pkspec1: The expression i cannot be used as a part specification.

Part::pkspec1: The expression i cannot be used as a part specification.

General::stop: Further output of Part::pkspec1 will be suppressed during this calculation.

NSum::nslim: Limit of summation -1. + 2. n is not a number.

NSum::nslim: Limit of summation -1. + 2. n is not a number.

NSum::nslim: Limit of summation -1. + 2. n is not a number.

General::stop: Further output of NSum::nslim will be suppressed during this calculation.

Do::iterb: Iterator {i, 1, dim} does not have appropriate bounds.

I don't where is that coming from, I mean to me it seems like the script itself have a problem. But I really don't think my professor would set me a wrong script. Could it be like in C language and I have to give kernel more ram ?

Progress! I suspect that the problem is in your command line input. Here is a simple example that gives the same error message:

C:\Users\chadk\temp>"\Program Files\Wolfram Research\Mathematica\11.1\WolframKernel.exe" -run "1+"

The kernel tries to evaluate ToExpression["1+"], and gives that error message.

Your input is missing the '<<'. Try this instead:

D:\ProgrammeFiles\mathematica\WolframKernel -run "<<double_chain.m" m

Or, if the double_chain file is in a different folder:

D:\ProgrammeFiles\mathematica\WolframKernel -run "Get[\"\\Path\\to\\the\\file\\double_chain.m\"]" m

where I am using escape keys on double quotes and backslashes.

I can understand your frustration! This is not trivial usage of the Wolfram kernel. I think that your instructor could have made this code much easier for students to use (for example, loading the package in a Mathematica notebook).

I'm not sure the best way to run the script at launch of Mathematica, and I wouldn't recommend that approach. As you have shown, you can load the file in Mathematica (using 'Get'), but you would need to change how the value of 'n' is specified in that file.

Using WolframKernel, you do not see a prompt because you passed the flag -noprompt. However, if you remove that flag, you'll have to change the code because 'n' will now be in the 4th position of \ $CommandLine (not the 5th). The simplest change to make would be to set 'n' to the last element in \$CommandLine. Now it will work regardless of the flags you pass to the kernel:

input=$CommandLine;
n=ToExpression[Last[input]];

I suspect that you just need to be more clear on the paths to the files. The input "<<doublechain.m" is telling the kernel to load the file "doublechain.m". If that file isn't in your current folder (or on your path), the kernel likely won't find it.

On my windows machine, I created a file (demo.m) in my temp folder. From my command prompt, I call run the file as follows:

C:\Users\chadk\temp>"\Program Files\Wolfram Research\Mathematica\11.1\wolfram.exe" -noprompt -run "<<demo.m" n

Alternatively, you could specify the path using ... -run "Get[\"\Users\chadk\temp\demo.m\"]" (where I have used escape characters for the folders as well as the String for the file name. ('<<' is synactic sugar for 'Get').