BUG: groupby sum turning `inf+inf` and `(-inf)+(-inf)` into `nan`
- [x] Closes #53606
- [x] [Tests added and passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests)
- [x] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit)
- [x] Added an entry in the latest `doc/source/whatsnew/v2.1.0.rst` file

This modifies `group_sum` in almost the way as #52964 for `group_mean`.
@Charlie-XIAO - small note, we like to use `BUG:` instead of `FIX`. See

https://pandas.pydata.org/pandas-docs/dev/development/contributing.html#finally-make-the-pull-request

for more details. > @Charlie-XIAO - small note, we like to use `BUG:` instead of `FIX`. See
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> https://pandas.pydata.org/pandas-docs/dev/development/contributing.html#finally-make-the-pull-request
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> for more details.

Oh sure, sorry about that. I mixed up with the prefix of another project lol. Thanks @Charlie-XIAO!