If you don't supply your finch with a new companion at the end of that week he/she may indeed appear to still be depressed. This is normal. Not so much that he/she is still mourning the loss, rather the finch is simply lonely. In time and if you are around more often the finch may perk up and appear happier. This happens when the finch starts to view you as part of the flock. A mirror in the cage can also help the finch cope with being alone but note that some aggressive species may feel threatened by their own reflection.