Arduino not running software

Hi

I have a arduino uno r3 below.

When I run something that uses the serial interface such as the AnalogInOutSerial example, it works perfectly with the tx led on.

My problem is that when I remove the usb cable from the arduino and reconnect it again, the program doesn't seem to be running anymore, tx remains off. It only starts operating again if I open the Serial Monitor.

If I use the Blink example, when I reconnect the cable it operates correctly so it seems to be a serial com problem.

Any Ideas of how to fix this problem?

I can't see what the problem is.
If the serial port is not connected to anything why should the chip send anything.
Note it is not the arduino that lights up the TX and RX LEDs but the USB / Serial chip.

Ok thanks, makes sense.