What is Shannon's source entropy
So,what exactly is Source entropy?


When the paper talks about "source entropy" all they mean is the entropy of the information source.  You can see that from the following passage in the paper- "Shannon showed that there must exist at least one encoding of the sequence generated by an information source which allows error-free transmission of the sequence when the channel capacity is larger than the source entropy."


  Now, problem_Set also mentions about source entropy but the formula is very different!


Yes, and no.  The formulas appear different, but they are not really.  The first formula is
$H_{source} = lim_{k→∞}\frac{1}{k}H_k$

$H_{source}$ is the total entropy of the source, and $H_k$ is the entropy of each possible discrete value that the random variables in $H$ can be.  Thus, it is almost a trivial statement that the total entropy is the sum of the individual value entropies, while the number of discrete values can go to $\infty$.

The problem set equation is-
$H(s) = \sum_i p_ilog_2\frac{1}{p_i}$

This is really the same equation, it just replaces the generic "$H_k$" with the implied $log_2\frac{1}{p_i}$.  The $\frac{1}{k}$ in the first equation is equivalent to the $p_i$ in the second equation- i.e. they are assuming that all of the discrete values of H are equally likely.  This is not generally true, but is true when the entropy is maximized.