- THREE CASES OF CONVERGENCE 
OF A POWER SERIES.
WE'VE SEEN EXAMPLES 
IN PREVIOUS VIDEOS
WHERE A POWER SERIES CONVERGES 
ON AN INTERVAL OF X VALUES,
SUCH AS THIS EXAMPLE.
THIS ONE CONVERGED 
ON THIS OPEN INTERVAL,
THIS ONE CONVERGED 
ON THIS INTERVAL
WHERE YOU GOT 15A 
TO CONVERGE THERE,
BUT DID NOT CONVERGE 
ON RIGHT END POINT.
AND ANOTHER EXAMPLE IN THE 
VIDEOS WAS THIS SERIES,
POWER SERIES,
AND IT CONVERGED 
ON THIS INTERVAL,
WHICH INCLUDE THE 
RIGHT END POINT,
BUT DID NOT INCLUDE THE LEFT.
SO WE SAW THAT TYPE OF EXAMPLE.
BUT CAN A POWER SERIES CONVERGE 
FOR ALL X VALUES,
ALL REAL NUMBERS?
OF COURSE, THAT WOULD MEAN
THAT THE RAYS OF CONVERGENCE IS 
INFINITY, SO CAN THAT HAPPEN?
SO HERE'S AN EXAMPLE, THIS NICE 
POWER SERIES RIGHT HERE.
IT EXPANDS OUT LOOKING 
LIKE THIS.
WE'LL DO THE RATIO TEST, 
DIVIDE, SIMPLIFY IT OUT,
THIS IS HOW THIS GOT TO BE 
THE NEXT STEP HERE,
WE COME DOWN HERE,
TAKE THE LIMIT AS N 
GOES TO INFINITY OF THIS,
OF COURSE, THE X SQUARED IS +N 
ANYWAY OR 0 AND FACTORS OUT.
AND THEN YOU JUST HAVE 
THIS LIMIT
AND THAT OF COURSE 
GOES TO THE 0,
SO X SQUARED x 0 = 0
AND THAT HAPPENS TO BE LESS 
THAN 1 FOR ALL X VALUES.
SO THE RAYS OF CONVERGENCE 
IS INFINITY
AND THE INTERVAL OF CONVERGENCE 
IS -INFINITY, LESS THAN X,
AND LESS THAN INFINITY,
IN OTHER WORDS, ALL REAL NUMBERS 
OR WE JUST SAY CONVERGES ON R.
SO IN OTHER WORDS, BACK TO HERE, 
NO MATTER WHAT X YOU PUT IN HERE
IT'S GOING TO MULTIPLY BY 0 
AND GET 0 AND 0 IS LESS THAN 1.
SO IT'S INTERESTING THAT, 
AND YOU'LL SEE WHY LATER,
THAT THIS SERIES HERE CONVERGES 
TO COSINE X FOR ALL X VALUES.
SO EVEN IF YOUR X WAS 20,000 
PI/3 YOU'D GET THE SAME VALUE
AS PLUGGING INTO THIS SERIES 
IF YOU TOOK A FAR ENOUGH OUT IN.
WHAT ABOUT THIS CASE?
CAN A POWER SERIES CONVERGE 
AT NO X VALUES
OR MAYBE ONE X VALUE?
WELL, LET'S TAKE THIS SERIES 
RIGHT HERE,
WHICH EXPANDS OUT LIKE THIS.
TAKE THE RATIO TEST IDEA, 
SIMPLIFIES DOWN TO THIS,
WHICH EQUALS THIS.
SO WHEN YOU TAKE THE LIMIT,
THE ABSOLUTE VALUE OF X FACTORS 
OUT AND THIS N +1,
THAT LIMIT GOES TO INFINITY NO 
MATTER WHAT X IS UNLESS X = 0.
IF X = 0 THEN WHAT YOU GET OUT 
THIS SERIES IS 1 + 0 + 00000
AND THAT ADDS IT 1, 
SO THE SERIES CONVERGES IT.
IN THAT CASE, IT EXCLUDES 0,
BUT IT DIVERGES FOR ALL OTHER 
X VALUES.
SO THE RAYS OF CONVERGENCE 
IS ZERO
BECAUSE IT DOESN'T CONVERGE 
FOR ANY OTHER VALUES
ABOUT THE CENTER 
OTHER THAN THE CENTER ITSELF,
WHICH HAPPENED TO BE ZERO 
IN THIS CASE.
