How to optimize an ORDER BY query using a range to avoid any filesorts

[Faury Rodriguez]

I have the following query:
SELECT
series_id, weight
from series_main
WHERE
series_main.db_id IN (1,3,4,5,6,7,8,9,10,11,12,13,32,33,34,35,36,37,38,40,41,42)
order by weight
limit 5000;

I also have the following indices:
PRIMARY (series_id, weight)
weight (weight)
db_weight (db_id, weight)

The explain output of the above query is:
id: 1
select_type: SIMPLE
table: series_main
type: range
possible_keys: db_weight
key: db_weight
key_len: 1
ref: NULL
rows: 1896002
Extra: Using where; Using filesort

The query uses the db_weight index to look up the db_ids but doesn't use the weight to sort the results.

How can I optimize the appropriate index or perhaps the query to get MySQL to use the index to find the the rows with a db_id in my range but use the same index to then sort the results by weight and avoid any use of filesort?