- HERE'S THE PROOF THAT 
THE CONVERGENCE
OF THE ABSOLUTE VALUES 
IMPLIES CONVERGENCE.
SO THE IDEA IS IF YOUR SERIES 
WITH ABSOLUTE VALUES
ON ALL THE TERMS,
IF THAT CONVERGES THEN SO DOES 
YOUR ORIGINAL SERIES
WITHOUT THE ABSOLUTE VALUES.
SO IN OTHER WORDS, THE ABSOLUTE 
VALUES ARE AT LEAST
AS BIG AS ALL YOUR "AN" VALUES.
SO IF THAT CONVERGES 
THEN SO DOES YOUR "AN."
SO HERE'S THE PROOF THAT YOU 
SHOULD KNOW FOR A TEST.
"AN" IS TRAPPED BETWEEN 
THE NEGATIVE ABSOLUTE VALUE
AND THE POSITIVE BECAUSE MAYBE 
THE "AN" TERMS ARE NEGATIVE,
YOU KNOW, WE DON'T KNOW.
SOME OF THEM COULD BE NEGATIVE, 
SOME COULD BE POSITIVE,
BUT IT'S DEFINITELY TRAPPED
IN-BETWEEN THESE TWO VALUES.
SO THEN WE ADD ABSOLUTE VALUE
OF "AN" ALL THROUGHOUT
THE INEQUALITY STATEMENT ON ALL 
3 PARTS, SO THIS GETS YOU TO 0,
IT ADDS IN HERE IN THE MIDDLE,
AND IT ADDS ON THE RIGHT 
AND YOU GET TWO OF THEM.
SO WE HAVE OUR GIVEN PART 
OF OUR STATEMENT:
IF THIS ABSOLUTE VALUE "AN" 
CONVERGES,
THEN CERTAINLY SO DOES 2 TIMES 
IT BECAUSE, YOU KNOW,
IF THE SERIES OF ABSOLUTE VALUES 
CONVERGES TO SOME NUMBER
THEN 2 x CONVERGES TO 2 x 
THAT NUMBER, SO IT CONVERGES.
NO PROBLEM THERE.
SO WE HAVE THAT 
AND THAT IS EQUAL TO, OF COURSE,
2 MULTIPLIED IN BECAUSE YOU 
COULD TAKE THE 2
AND FACTOR IT BACK IN.
ALL RIGHT, SO WE HAVE THAT THIS 
SERIES OF "AN"
+ ABSOLUTE VALUE OF N CONVERGES.
SINCE WE KNOW THIS ONE DOES, 
DUE TO OUR GIVEN STATEMENT,
THAT CONVERGES AND THE TERMS OF 
OUR SERIES HERE ARE LESS THAN
OR EQUAL TO THOSE TERMS 2 
x THE ABSOLUTE VALUE OF "AN."
SO BY THE COMPARISON OF SERIES,
SINCE WE HAVE THAT THIS SERIES 
CONVERGES
BECAUSE EACH OF ITS TERMS ARE 
LESS THAN OR EQUAL TO SOMETHING
THAT CONVERGES.
NOW, STEP FIVE.
SINCE "AN" IS LESS THAN OR EQUAL 
TO "AN" + ABSOLUTE OF N,
LET'S TALK ABOUT 
THAT FOR A SECOND.
IF YOU ADD ON "AN" ABSOLUTE 
VALUE,
THAT'S A POSITIVE NUMBER OR A 0 
THAT YOU'RE ADDING ON TO "AN."
SO, OF COURSE, THE "ANs" 
ARE LESS THAN OR EQUAL TO THAT,
SO THAT MEANS THAT THIS SERIES 
IS LESS THAN OR EQUAL
TO THIS SERIES 
BY A TERM BY TERM COMPARISON.
WELL, WE ALREADY SAID IN STEP 
FOUR THAT THIS SERIES CONVERGES,
SO BY COMPARISON THAT MEANS 
THE SUM OF "AN" CONVERGES.
SO THAT PROVES THE THEOREM 
BECAUSE BACK UP HERE
IN THE STATEMENT WE WERE SAYING 
IF THE ABSOLUTE VALUE
OF THE TERMS, IF THAT SERIES 
ENDED UP CONVERGING,
THEN SO DOES THE ORIGINAL.
SO WE ASSUMED THAT THE ABSOLUTE 
VALUE TERMS SERIES,
WE ASSUME THAT THAT CONVERGED
AND THAT LED US ALL THE WAY 
DOWN HERE
TO THE SUM OF "AN" CONVERGING 
AND THAT PROVES THE THEOREM.
