B757-200 AT FL360 WITH ZME EXPERIENCED TCAS RA WITH UNREPORTED TFC; ATC CONFLICTING TFC FAILED TO DESCEND.

Narrative

I WAS THE PLT MONITORING ON FLT ABC ATL-ZZZ. OUR POS WAS JUST NW (MAYBE 10 MI OR SO) OF RQZ VOR (HUNTSVILLE; AL) ON THE WAY TO JKS VOR (LEXINGTON-PARSONS; TN) AT FL360. IMMEDIATELY AFTER REACHING CRUISE; I BEGAN AN INFORMATIVE PA TO OUR PAX WHEN WE RECEIVED A TA FROM OUR TCAS. I IMMEDIATELY STOPPED THE PA AND LOOKED AT THE DISPLAY. THE YELLOW TARGET WAS AT OUR APPROX 11: 30 O'CLOCK POS AND CLOSING AT OUR ALT. I THEN ATTEMPTED TO OBTAIN VISUAL CONTACT WITH THE ACFT. AS SOON AS I LOOKED OUT THE COCKPIT WINDOW; THE TA CHANGED TO AN RA AND COMMANDED 'CLB...CLB.' I LOOKED AT THE IVSI AND IT COMMANDED APPROX 1200 FPM CLB. THE CAPT IMMEDIATELY AND POSITIVELY STARTED A CLB THAT PUT THE RATE OF CLB IN THE GREEN BAND ON THE IVSI. I ADVISED ZME THAT WE HAD AN RA AND THAT WE WERE CLBING. WE CLBED TO FL370 WHEN THE TCAS SAID 'CLR OF CONFLICT.' I LOOKED AT THE DISPLAY AND IT SAID 'TA BEHIND.' I NEVER SAW THE OTHER ACFT. THE CAPT SAID HE DID SEE IT BEFORE IT WENT UNDER US. I THEN INFORMED ATC THAT WE WERE RETURNING TO FL360. ZME RESPONDED THAT THE TFC WAS A CESSNA CITATION UNDER ATLANTA ARTCC CTL THAT FAILED TO DSND WHEN TOLD. WHY THESE 2 ARTCC'S LET THESE 2 AIRPLANES GET THIS CLOSE WITHOUT WARNING; I DO NOT UNDERSTAND. THE TCAS SYS WORKED AS ADVERTISED.