versäumen
to miss
To fail to attend or reach an opportunity, appointment, or means of transport in time.
Ich habe die letzte Bahn versäumt und musste ein Taxi nehmen.
I missed the last train and had to take a taxi.

to miss / neglect
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to miss
To fail to attend or reach an opportunity, appointment, or means of transport in time.
Ich habe die letzte Bahn versäumt und musste ein Taxi nehmen.
I missed the last train and had to take a taxi.

The verb 'versäumen' (to miss / neglect) is inseparable. The prefix 'ver-' remains firmly attached to the stem 'säumen' in all sentence forms and tenses.
Er hat den Anfang des Films versäumt.
He missed the beginning of the film.
When 'versäumen' (to neglect) describes an omitted action, it is often used with an infinitive construction with 'zu'. The formal variant is 'es versäumen, etwas zu tun' (to neglect to do something).
Versäumen Sie nicht, Ihre Rechnungen pünktlich zu bezahlen.
Do not neglect to pay your bills on time.
In this meaning, 'versäumen' (to miss) is often combined with means of transport, appointments, or opportunities. Typical examples are: 'den Zug/Bus versäumen' (to miss the train/bus), 'einen Termin versäumen' (to miss an appointment), or 'eine Gelegenheit versäumen' (to miss an opportunity).
Wenn du jetzt nicht gehst, versäumst du deinen Flug.
If you don't leave now, you'll miss your flight.
'Versäumen' (to miss / neglect) belongs more to elevated or formal language. In everyday speech, the verb 'verpassen' (to miss) is often used, especially for appointments or means of transport.
Der Minister versäumte es, die Öffentlichkeit rechtzeitig zu informieren.
The minister neglected to inform the public in time.
to forget
'vergessen' and 'versäumen' can both mean that you do not do something. 'vergessen' emphasizes that you did not think of it, for example a key or a task. 'versäumen' fits especially well with appointments, opportunities, deadlines, or means of transport. It often sounds a bit more formal. Learners may confuse the words because you can say that you 'vergessen' something or 'versäumt' something. In practice: if the reason is in your head, 'vergessen' is very natural. If you do not take advantage of an opportunity or do not attend an appointment, 'versäumen' is more precise.

Sie versäumt den Termin im Krankenhaus, weil der Bus zu spät kommt.
She misses the appointment at the hospital because the bus arrives too late.
Nach dem Gespräch auf dem Amt merke ich, dass ich meine Unterlagen vergessen habe.
After the conversation at the government office, I realize that I forgot my documents.