- PARTIAL SUMS 
AND CONVERGENCE OF A SERIES.
YOU CAN READ THIS PART 
RIGHT HERE: THE PARTIAL SUMS
ARE S SUB N, AND WE'LL TALK 
ABOUT WHAT THOSE ARE,
AND IF THEY LIMIT 
OUT TO THE NUMBER CAPITAL S,
SO THAT AS N GOES TO INFINITY, 
THE PARTIAL SUMS CONVERGE
TO S, THEN THE SERIES CONVERGES, 
AND IT CONVERGES TO THAT S.
SO YOU CAN WRITE THE SERIES 
SUM = S.
IF S SUB N DOES NOT EXIST BY WAY 
OF EITHER GOING TO INFINITY
OR FLIP FLOPPING BACK 
AND FORTH, SOMETHING LIKE THAT,
THAN WE SAY THE SERIES 
DIVERGES.
SO HERE'S AN EXAMPLE,
HERE'S A SEQUENCE 1, 1/2, 1/4, 
1/8, CONTINUING IN THAT PATTERN.
SO THE SEQUENCE 
IS THE LIST OF NUMBERS.
THE SERIES IS THE SUM 
OF ALL THE SEQUENCE TERMS
FROM THE FIRST ONE 
TO INFINITY AND THAT JUST
IS DENOTED THIS WAY.
SO IT'D BE THE SUM OF ALL 
THESE SEQUENCED TERMS
ON TO INFINITY, 
SO IT'S AN INFINITE SUM.
A PARTIAL SUM IS A SUM 
OF THE FIRST N TERMS
OF THE SEQUENCE, 
SO IT'S A FINITE SUM,
YOU STOP AT "A" SUB N 
AND THAT'S WHY, YOU KNOW,
I HAD TO WRITE THIS WITH 
THE "I" COUNTER INSTEAD
OF THE N BECAUSE 
THE N IS THE LAST ONE,
AND THAT'S CALLED S SUB N, 
THAT'S A PARTIAL SUM.
SO THE OTHER NOTATION 
IS THAT IF THIS NUMBER,
THE LIMIT AS THE PARTIAL SUMS OF 
N GOES TO INFINITY EQUALS S,
THEN SINCE THOSE PARTIAL SUMS 
ARE EQUAL TO THESE, YOU KNOW
WITH THE LIMIT, SO THE S SUB N, 
EACH ONE OF THOSE IS EQUAL
TO THE SUM, STOPPING 
AT N IS THE PARTIAL SUM,
AND YOU TAKE THE LIMIT AS N GOES 
TO INFINITY OF THOSE,
THAT'S WHAT IT'S MEANT 
BY HAVING AN INFINITY UP HERE.
THAT'S WHAT THAT MEANS.
OKAY, SO IF THAT PARTIAL SUM 
EXISTS THEN THE SERIES CONVERGES
AND CONVERGES TO THE LIMIT.
SO HERE'S YOUR EXAMPLE.
THE SEQUENCE TERMS ARE A LIST 
OF TERMS, THE PARTIAL SUMS,
S SUB 1, IS A SUM 
OF THE FIRST ONE.
WELL, THE FIRST ONE IS JUST ONE, 
SO THAT'S ONE.
S2 IS THE SUM OF THE FIRST 
2, 1 + 1/2 WHICH IS 3/2.
S3 IS THE SUM OF THE FIRST 
3 LIKE THIS, THAT ADDS UP
TO 7/4 AND THE SUM OF THE 1st 4 
TERMS EQUALS ALL THIS,
EQUALS 15/8.
NOW, ONE METHOD TO FIGURE 
OUT THE SERIES SUM
IS TO FIGURE OUT THE PARTIAL 
SUM PATTERN.
IN OTHER WORDS, 
WHAT IS THE Nth PARTIAL SUM?
SO YOU TAKE THESE NUMBERS 
AND YOU SAY WELL,
IN THE NUMERATOR, 
IT'S 1, 3, 7, 15.
SO I WROTE THOSE DOWN 
HERE; 1, 3, 7, 15.
I KNOW THERE'S SOME KIND OF 
POWER OF 2 GOING ON IN HERE,
SO I ANALYZE THE DIFFERENCES, 
3 - 1 = 2,
7 - 3 = 4, 15 - 7 = 8,
SO SURE ENOUGH THERE IS A POWER 
OF 2 PATTERN
FROM ONE OF THE NUMERATOR VALUES 
TO THE NEXT.
SO I TAKE A STAB 
AT IT AND I JUST SAY OKAY,
WELL, I THINK THE NUMERATOR 
NUMBERS ARE 2 TO THE N, OKAY.
WELL, THAT'S NOT RIGHT BECAUSE 
2 TO THE 1 IS 2 NOT 1,
BUT WHAT IF I SUBTRACT THE 1?
BECAUSE LOOK AT 1 = 2 - 1, 
3 = 4 - 1, 7 = 8 - 1,
15 = 16 - 1, SO THEN I HAVE 
THIS NUMERATOR PATTERN.
SO WHAT YOU'RE DOING 
IS YOU'RE GUESSING,
ADJUSTING IT, AND THEN CHECKING 
IT TO MAKE SURE
ALL THE NUMBERS WORK.
NOW WE'RE GOING TO GO 
FOR THE DENOMINATOR PATTERN.
THAT'S 1, 2, 4, AND 8.
SO THOSE ARE DIRECT 
POWERS OF 2.
SO I WOULD SAY 2 
TO THE N, BUT FOR N = 1,
2 TO THE 1st IS 2 NOT 1, 
SO I'M OFF BY 1.
SO I MUST EITHER GO N + 1 
OR N - 1, SO WE GO N - 1.
SO THE N = 1, 2 TO THE 0 = 1, 
CHECK, SO THIS IS WORKING.
SO THIS IS YOUR PARTIAL 
Nth SUM AND WE JUST WANT
TO FIND THE LIMIT 
AS N GOES TO INFINITY.
SO I THOUGHT OF BREAKING IT UP 
IN THE FRACTION,
DIVIDING THE DENOMINATOR INTO 
BOTH OF THE NUMERATORS
WITH A SUBTRACTION SIGN 
IN THERE.
2 TO THE N DIVIDED 
BY 2 TO THE N - 1,
THAT'D BE LIKE 2 TO THE 4th/2 
TO THE 3rd IS 2
OR 2 TO THE 7th/2 
TO THE 6th IS 2.
SO THIS JUST AUTOMATICALLY 
CANCELS OUT TO 2,
THERE'S NO LIMITING GOING 
ON THERE.
THEN ON THIS ONE, N GOES 
TO INFINITY, THIS DENOMINATOR
GOES TO INFINITY, 1/INFINITY 
GOES INTO 0,
SO THERE THE LIMIT IS IN EFFECT 
IN THAT IT'S EQUAL TO 2.
SO ALL THOSE ALL NUMBERS 
IN THE SEQUENCE ADD UP TO 2.
SO THE PARTIAL SUMS 
GO TO 2 AS N GOES TO INFINITY,
HENCE THE SERIES, WHICH IS THE 
LIMIT OF THE PARTIAL SUMS,
GOES TO 2.
WE CAN CHECK THAT BECAUSE 
THIS PARTICULAR SEQUENCE
HAPPENS TO BE A GEOMETRIC 
SEQUENCE.
SO THEY'RE NOT ALWAYS GEOMETRIC, 
BUT THIS ONE IS.
IT TAKES ON THIS FORM HERE, 
NOTICE THAT IT'S GEOMETRIC.
YOU CAN PUT INTO THIS FORM 
SOME CONSTANT NUMBER
OUT IN THE FRONT, 
SO THAT'S THE FIRST TERM.
THAT'S ONE.
THE RATIO THAT'S BEING HIT 
ON EACH TERM TO GET
TO THE NEXT ONE IS 1/2 AND SO 
YOU HAVE 1/2 TO THE N - 1
IS YOUR SEQUENCE TERMS FORM 
AND THEN WE SAW PREVIOUSLY
IN ANOTHER SECTION THAT THAT 
GEOMETRIC SERIES ADDED UP
TO THIS QUANTITY HERE, 
SO "A" = 1 AND R = 1/2.
SO WHEN YOU TAKE THE LIMIT 
AS N GOES TO INFINITY,
YOU'RE GOING TO HAVE 
1/2 TO N = 0,
SO THAT'S 1 - 0/1 - 1/2 = 2, 
SO THAT CHECKS OUT.
