An LR45 Captain reported an NMAC at FL410 when an aircraft at FL400 climbed toward them in response to an RA.

Narrative

We were maintaining FL410 as the cruise altitude and had just completed lateral deviations around a line of thunderstorms. Shortly after clearing the weather we received a TA for traffic at approximately 10 o'clock; 900 feet below our altitude and climbing. We tried to pick up the aircraft visually and I commented 'Let's be ready for an RA.' About 10 seconds after the TA we received an RA to climb. The pilot flying responded quickly and began the climb. I told ATC; 'Center; we have received an RA for an aircraft below us; we are responding'. Immediately after our transmission the other aircraft told ATC 'We have the other aircraft in sight. We are also responding to a climb RA.' ATC responded '[callsign]; confirm you are maintaining assigned altitude FL400?' The other aircraft responded they were and again stated they had received an RA to climb. There was then a short pause on the radio. The closest point we noted the other aircraft on the TCAS display was 400 feet below us. After the TCAS stated 'Clear of conflict' we returned to FL410. We double-checked the altimeters were set at 29.92. Our earlier ground and RVSM altimeter checks had been good. We were confident we had been where we were supposed to be. ATC then asked the other aircraft several times to confirm they had been maintaining FL400 and they responded they had been. The Controller asked us to confirm we had been at FL410 and received a climb RA. I responded that was affirmative. We were then given a frequency change to the next sector. I have experienced a number of RAs at low altitude; especially near busy airports; but I've never had an RA at high altitude. I was amazed how fast the situation progressed after the initial TA. The high closure rates in this event impressed upon me the importance of immediate response to an RA.