BackgroundServiceOnObservableSchedule.cs

    public class BackgroundServiceOnObservableSchedule : BackgroundService
    {
        private static readonly ILogger Logger = Log.ForContext<BackgroundServiceOnObservableSchedule>();
        
        private readonly IObservable<Unit> workerSequence;
        private IDisposable workerSubscription;

        public BackgroundServiceOnObservableSchedule()
        {
            workerSequence = Observable
                .Timer(TimeSpan.FromSeconds(5))
                .Do(_ => Logger.Information("Doing the work ..."))
                .Select(_ => DoTheWork())
                .RetryWhen(errors => errors.SelectMany(ex =>
                {
                    // This place is to filter on exception and log/throw/continue
                    Logger.Error(ex, $"Failure in {nameof(BackgroundServiceOnObservableSchedule)}");
                    return Observable.Return(Unit.Default); 
                }))
                .Repeat();
        }

        private Unit DoTheWork() => Unit.Default;

        protected override Task ExecuteAsync(CancellationToken stoppingToken)
        {
            workerSubscription = workerSequence.Subscribe(_ => { });
            
            return Task.CompletedTask;
        }

        public override void Dispose()
        {
            workerSubscription.Dispose();
            base.Dispose();
        }
    }