gh-39480: Added enhancement for power series that allows access to the coefficients of specific terms
    
Fixes #39314.  Adds an enhancement to power series that when given a
tuple of values corresponding to the powers of the term gives you the
coefficient for said term. Functionality is similar to how coefficients
are retrieved for multivariable polynomials.  Also added a test for the
enhancement.

### 📝 Checklist

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- [X] The title is concise and informative.
- [X] The description explains in detail what this PR is about.
- [X] I have linked a relevant issue or discussion.
- [X] I have created tests covering the changes.
- [X] I have updated the documentation and checked the documentation
preview.

### ⌛ Dependencies
    
URL: #39480
Reported by: Noel-Roemmele
Reviewer(s): Martin Rubey, Noel-Roemmele, Travis Scrimshaw

Author: Release Manager

src/sage/rings/multi_power_series_ring_element.py

@@ -673,7 +673,11 @@ def _im_gens_(self, codomain, im_gens, base_map=None):
 
     def __getitem__(self, n):
         """
-        Return summand of total degree ``n``.
+        Return the coefficient of the monomial ``x1^e1 * x2^e2 * ... * xk^ek``
+        if ``n = (e_1, e2, ..., ek)`` is a tuple whose length is the number of
+        variables ``x1,x2,...,xk`` in the power series ring.
+
+        Return the sum of the monomials of degree ``n`` if ``n`` is an integer.
 
         TESTS::
 
@@ -690,9 +694,30 @@ def __getitem__(self, n):
             ...
             IndexError: Cannot return terms of total degree greater than or
             equal to precision of self.
-        """
+
+        Ensure that the enhancement detailed in :issue:`39314` works as intended::
+
+            sage: R.<x,y> = QQ[[]]
+            sage: ((x+y)^3)[2,1]
+            3
+            sage: f = 1/(1 + x + y)
+            sage: f[2,5]
+            -21
+            sage: f[0,30]
+            Traceback (most recent call last):
+            ...
+            IndexError: Cannot return the coefficients of terms of total degree
+            greater than or equal to precision of self.
+        """
+        if type(n) is tuple:
+            if sum(n) >= self.prec():
+                raise IndexError("Cannot return the coefficients of terms of " +
+                                 "total degree greater than or equal to " +
+                                 "precision of self.")
+            return self._bg_value[sum(n)][n]
         if n >= self.prec():
-            raise IndexError("Cannot return terms of total degree greater than or equal to precision of self.")
+            raise IndexError("Cannot return terms of total degree greater " +
+                             "than or equal to precision of self.")
         return self.parent(self._bg_value[n])
 
     def __invert__(self):