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Card 13 / 13:
Taylor’s theorem with remainder
for a function f and the n th Taylor polynomial for f at x = a , the remainder R n ( x ) = f ( x ) − p n ( x ) satisfies R n ( x ) = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x − a ) n + 1 for some c between x and a ; if there exists an interval I containing a and a real number M such that | f ( n + 1 ) ( x ) | ≤ M for all x in I , then | R n ( x ) | ≤ M ( n + 1 ) ! | x − a | n + 1
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