- HERE'S AN EXAMPLE OF EMPLOYING 
THE RATIO TEXT TO TEST A SERIES
RIGHT HERE FOR CONVERGENCE.
SO, OF COURSE, 
WE HAVE THE FACTORIAL,
AND A FACTORIAL IS A GOOD SIGN 
OF WHEN YOU'RE GOING TO BE USING
THE RATIO TEST.
THERE'S OTHER CASES TOO, 
OF COURSE,
BUT FACTORIAL IS A COMMON ONE TO 
EMPLOY THE RATIO TEST WITH.
SO YOUR An + 1, THE NEXT TERM,
YOU GOT TO BE REALLY CAREFUL 
ABOUT THIS,
THE n IS REPLACED BY n + 1.
SO YOU DON'T SAY n! + 1, YOU SAY 
((n + 1)!)
AND THAT'S BEING SQUARED.
THEN YOU REPLACE THIS n, 
YOU GOT TO BE VERY CAREFUL,
WITH (n + 1) IN THE PARENTHESES.
IT'S NOT THE SAME THING AS 2n + 
1, SO IT'S NOT 2n + 1,
IT'S NOT THAT, 
ENDS UP BEING 2n + 2. OKAY?
AND THE An IS JUST THE ORIGINAL 
TERM OF THE SERIES.
SO AFTER YOU PUT IN THAT n + 1 
CAREFULLY IN PLACE OF n,
THEN YOU FORM YOUR RATIO 
OF ABSOLUTE VALUES,
AND IT'S WRITTEN DOWN HERE.
AND THEN SOMETIMES THE ABSOLUTE 
VALUES ARE CRUCIAL,
SOMETIMES THEY'RE NOT.
THESE ARE FACTORIALS. THOSE 
ARE POSITIVE NUMBERS.
THE n IS STARTING AT 1.
SO THAT'S ALL POSITIVE NUMBERS 
IN THERE,
ALL POSITIVE NUMBERS IN HERE
BECAUSE THE FACTORIAL 
IS POSITIVE,
SO YOU DON'T NEED THE ABSOLUTE 
VALUES ANYMORE
GOING TO STEP 2 FOR THIS CASE.
SO I TOOK THE ABSOLUTE VALUES 
OFF.
I ALSO INVERTED THE DENOMINATOR 
AND MULTIPLIED BY IT OVER HERE,
SO IT'S JUST MULTIPLICATION 
OF FRACTIONS. OKAY?
THEN I KNEW THAT I WAS GOING 
TO HAVE TO CANCEL OUT
WITH ONE OF THESE n FACTORIALS, 
SO I TOOK THE (n + 1)!
RIGHT HERE, AND I BURNED OFF ONE 
OF THE FACTORS HERE.
BECAUSE (n + 1)!, 
(n + 1)! = (n + 1)(n)(n-1),
DOT, DOT, DOT, DOT, 
ALL THE WAY DOWN TO 1.
THAT'S AN (n + 1).
SO I TOOK THE (n + 1) 
AND I WROTE THIS AS n!
AND THAT'S HOW I GOT 
((n + 1) n!)
AND THAT'S BEING SQUARED.
THEN HERE, I JUST DISTRIBUTED 
THE 2n TO HERE ALGEBRAICALLY,
BECAUSE I AM LOOKING AHEAD TO 
CANCELING OUT WITH THIS (2n)!,
BUT IT'S NOT QUITE READY YET.
THEN I HAVE THE SQUARE ON THIS 
PRODUCT,
SO I CAN JUST COME OVER HERE.
I PUT THE SQUARE ON BOTH OF THE 
FACTORS, THAT'S ALGEBRA.
I BURNED OFF A COUPLE OF TERMS 
OF THIS FACTORIAL.
IT'S (2n + 2), IT STARTS WITH.
SUBTRACTING 1 FROM THAT IS ONLY 
(2n + 1),
GOING DOWN ONE INTEGER 
AT A TIME,
AND THEN YOU'RE LEFT WITH THE 
WHOLE REMAINDER, WHICH IS (2n)!
BECAUSE YOU HAD (2n + 2)(2n + 1)
(2n + 0), THAT'S (2n)!,
AND THEN I LEFT THIS PART THE 
WAY IT WAS.
THEN I SEE THAT I COULD JUST 
CANCEL OUT THE (2n)!.
LET'S DO THAT AGAIN RIGHT HERE, 
(2n)!, (2n)!, THE (n!) SQUARED,
AND THE (n!) SQUARED, 
AND I'M LEFT WITH THIS.
SO NOW AT LEAST I GOT RID 
OF THOSE FACTORIALS,
WHICH IS WHAT THE GOAL 
IS REALLY.
SO THEN I'M DOWN TO HERE.
I CANCELLED OUT THE SQUARE 
ON THIS
WITH THE ONE POWER DOWN HERE
AND JUST GOT A SINGLE 
n + 1 QUANTITY.
THEN THAT CANCELLED OUT,
SO I WAS LEFT WITH THIS ON THE 
BOTTOM.
AND THEN I MOVED THE 2n, 
DISTRIBUTED n, AND GOT THIS.
NOW I'M SUPPOSED TO BE TAKING 
THE LIMIT OF THAT RATIO
AS n GOES TO INFINITY, 
SO I DID THAT HERE
AND THIS IS WHAT IT 
CAME OUT TO BE.
AS YOU CAN SEE, AS n GOES TO 
INFINITY,
THAT 1 AND 2 DOESN'T MATTER.
WE USE OUR NORMAL TECHNIQUES 
AND WE GET 1/4.
THIS IS GOING TO GO TO 1/4.
AND 1/4 IS LESS THAN 1, SO WE'RE 
IN CASE 1 WHERE THAT APPLIES,
AND SO THAT'S THE CASE WHERE THE 
SERIES CONVERGES.
NOW YOU MIGHT BE THINKING, 
"CONVERGES TO WHAT?"
WELL, THAT'S NOT PART OF THE 
ORIGINAL QUESTION,
BUT WE CAN STILL ANSWER IT.
CONVERGES TO WHAT?
IT DOES NOT CONVERGE TO 1/4.
NO, NOT 1/4. THAT WAS JUST 
THE LIMIT OF THE RATIOS.
THAT'S NOT WHAT THIS SERIES SUM 
ADDS UP TO.
IT'S A COMPLETELY 
DIFFERENT THING.
WE WANT TO KNOW WHAT THE SUM 
ADDS UP TO,
SO WE USE OUR RIEMANN 
SUM PROGRAM.
SO IN Y1, YOU PUT IN THIS 
AND YOU DON'T NEED THE n.
YOU'RE IN REGULAR MODE WITH THE 
X FACTORIAL SQUARED (2X)!.
NOW YOU MIGHT BE THINKING,
"OH, I'M WORRIED ABOUT X BEING 
NOT AN INTEGER,"
BUT WHEN WE RUN THE RIEMANN 
PROGRAM,
WE'RE JUST GOING TO RUN IT WITH 
INTEGERS THE WAY WE SET IT UP,
SO IT WILL ALL WORK OUT DEFINED.
IF YOU DO SOMETHING WRONG LATER 
WITH RUNNING THIS PROGRAM,
IT WILL GRIPE BECAUSE IT DOESN'T 
KNOW WHAT 3/2 FACTORIAL,
FOR EXAMPLE, IS.
SO TO GET THE FACTORIAL SYMBOL, 
YOU GO TO THE MATH KEY,
THEN YOU ARROW ACROSS TO 
PROBABILITY
AND GO TO NUMBER 4 ON THIS, 
THIS NUMBER 4 FOR THIS ONE.
THAT'S YOUR FACTORIAL SIGN,
AND YOU ENTER THAT IN AND THAT'S 
HOW YOU GET YOUR FACTORIAL.
SO THEN YOU PUT YOUR SERIES TERM 
RIGHT THERE.
AND SUPPOSE YOU WANT TO
RUN 1 TO 10.
NOW WHY IS IT ONLY 10,
BECAUSE YOU'RE DEALING WITH 10 
FACTORIAL IN THERE.
YOUR CALCULATOR IS GOING TO BE 
CRUNCHING ALONG FOR A LONG TIME
IF YOU TRY TO PUT IN 
SAY A HUNDRED UP HERE.
10 IS BIG. AND EVEN ON A 
CALCULATOR, 10 MIGHT BE BIG,
SO YOU'LL JUST 
HAVE TO EXPERIMENT.
SO LET'S SAY YOU WANTED THAT 10.
THEN YOU'D BE, THE WAY I SHOWED 
YOU IN ANOTHER VIDEO,
YOU'D BE GOING FOR A LEFT.
AND SO YOU'RE OVER HERE WITH 
THIS PICTURE.
THESE ARE A LEFT SUM PICTURE 
RIGHT HERE.
SO YOU SEE TO GET TO FULL 10 
VALUES,
YOU WANT TO HAVE 11.
YOU WANT TO END IN 11 AND START 
AT 1, BUT YOU NEED 10 STEPS.
BECAUSE WHATEVER YOU DO 
WITH THIS,
YOU BETTER MAKE SURE THAT
B - A/n, 11 - 1/10 TURNS OUT TO 
BE 1. THAT HAS TO BE A 1,
BECAUSE YOUR DELTA HAS TO BE 1
SO THAT THE HEIGHTS OF THESE 
RECTANGLES
ARE THE ACTUAL An VALUES. 
OKAY?
SO YOU RUN YOUR RIEMANN PROGRAM 
AND NOT THE INTEGRAL,
BECAUSE THE INTEGRAL AGAIN,
WELL, WE'LL EXPLAIN IT IN A 
SECOND.
SO YOU LOOK AT THE LEFT, ANSWER, 
AND IT COMES OUT .736.
AND YOU PROBABLY HAD SOME OTHER 
DIGITS IN YOUR CASE,
BUT MINE ROUNDED OFF TO .736
WHEN I DID IT ON THE EQUIVALENT 
ON THE COMPUTER.
OKAY. SO WHY RUN THE RIEMANN 
PROGRAM
AND NOT THE INTEGRAL PROGRAM?
THAT IS BECAUSE RIEMANN ONLY 
COMPUTES LEFT AND RIGHT.
IT DOESN'T DO MIDPOINT 
OR SIMPSON.
IF IT WAS TRYING TO CALCULATE 
THE MIDPOINT,
IT WOULD TRY TO CALCULATE, 
FOR EXAMPLE, IN BETWEEN 1 AND 2,
IT WOULD BE 3/2.
IT WOULD PUT (3/2)!,
AND IT WOULD STOP RIGHT THERE 
AND GRIPE
BECAUSE THAT WOULD BE UNDEFINED.
FACTORIALS ARE DEFINED ON THE 0, 
1, 2, 3 POSITIVE INTEGERS + 0.
0! IS 1, 1! IS 1, 2! 
IS 2 x 1, WHICH IS 2,
3! IS 3 x 2 x 1, WHICH IS 6, 
AND SO ON.
SO RECAPPING, WE WANTED TO FIND 
OUT THE SERIES SUM
AND THAT SERIES SUM WAS RIGHT 
HERE, .736.
BE INTERESTING TO TRY LOWER 
NUMBERS IN 10 TO SEE HOW CLOSE
YOU WOULD GET TO .736.
AND IF YOUR CALCULATOR IS 
RUNNING REALLY SLOW, YOU KNOW,
IF YOU PUSH THIS UP TO 15,
YOU KNOW, YOU COULD SEE 
HOW SLOW IT WOULD TAKE
AND IF YOU GET ANY MORE 
ACCURACY.
