And that sketch does blink if the modules TX pin is connected to pin 10 on the Arduino but it doesn't blink if it's connected to pin 0? If that's the case I would connect my scope and see how the signals look like.
And that sketch does blink if the modules TX pin is connected to pin 10 on the Arduino but it doesn't blink if it's connected to pin 0? If that's the case I would connect my scope and see how the signals look like.