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.

I can't tell at which point in your code these errors are coming from.
The error messages indicate that there is a ToExpression wrapped around a string. Here is an input that gives the same error:

In[6]:= ToExpression["\U"]
During evaluation of In[6]:= Syntax::stresc: Unknown string escape \U.
Out[6]= $Failed

So perhaps this is in your -run "Get[ path to file]" code??? On windows machines, you have to be very careful with the path names because "\" is an escape character in Mathematica. A safer way to specify the path is to use FilenameJoin, e.g.,

In[7]:= FileNameJoin[{"Users","chadk","temp","demo.m"}]
Out[7]= Users\chadk\temp\demo.m

Then your command line input would look something like this:

C:\Users\chadk> "\Program Files\Wolfram Research\Mathematica\11.1\WolframKernel.exe" -run "Get[FileNameJoin[{\"Users\",\"chadk\",\"temp\",\"demo.m\"}]]" n

where the escape character (/) is used for the strings inside the FileName

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 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').