Dear questioner, thank you very much for your important question.

Fasting six days of Shawwal after the obligatory fast of Ramadan is a recommendable action and not obligatory. It is desirable for the Muslim to fast six days of Shawwal, which involves great virtues and an immense reward. Whoever fasts these six days will have recorded for him a reward as if he had fasted a whole year, as was reported in an authentic hadith from the Prophet (peace be upon him). Abu Ayyub (may Allah be pleased with him) reported that the Messenger of Allah (peace be upon him) said: “Whoever fasts Ramadan and follows it with six days of Shawwal, it will be as if he fasted for a lifetime.” (Muslim)

The Prophet (peace be upon him) explained this when he said: “Whoever fasts for six days after `Eid al-Fitr has completed the year: (whoever does a good deed (hasanah) will have ten hasanat like it).”

According to another report: “Allah has made for each hasanah ten like it, so a month is like fasting ten months, and fasting six days completes the year.” (An-Nisaa’i)

It was also narrated by Ibn Khuzaymah with the wording: “Fasting for the month of Ramadan brings the reward of ten like it, and fasting for six days brings the reward of two months, and that is the fasting of the whole year.”

Moreover, fasting six days of Shawwal makes up for any shortfall in a person's obligatory Ramadan fasts, because no one is free of shortcomings or sins that have a negative effect on his fasting.

On the Day of Resurrection, some of his supererogatory deeds will be taken to make up the shortcomings in his obligatory deeds, as the Prophet (peace be upon him) said: “The first thing for which people will be brought to account on the Day of Resurrection will be their prayer. Our Lord, may He be glorified and exalted, will say to His angels – although He knows best – ‘Look at the prayer of My slave, whether it is complete or incomplete.’ If it is perfect, it will be recorded as perfect, and if something is lacking, He will say, ‘Look and see whether My slave did any voluntary prayers.’ If he did some voluntary prayers, [Allah] will say, Complete the obligatory actions of My slave from his voluntary actions.’ Then all his actions will be dealt with in a similar manner.” (Abu Dawud).

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Source: Taken with modifications from Islamqa.info by Sheikh Muhammed Salih Al-Munajjid