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The Taylor Series

Problems

The Remainder Term

Some Common Taylor Series

Problem : Approximate e 0.1 to within 10-3 using Taylor's formula.

The error term in Taylor's formula is

r n(x) = x n = x n < x n    

where c is some number in the interval (0, 0.1) . Thus r 3(0.1) < 1/3000 < 10-3 , so it is sufficient to go up to the degree 2 term in approximating e 0.1 . We have

e 0.1 1 + 0.1 + = 1.105    

Problem : Approximate sin(- 0.1) to within 10-5 . The Taylor series for sin(x) begins

x - + - + ...    

The absolute value of the error term in Taylor's formula is

| r n(x)| = x n = | x|n    

Noting that | r 4(- 0.1)|≤| 0.1|4/4!≤10-5 , we see that we need to use up to the degree 4 term in the Taylor series. We have

sin(- 0.1) (- 0.1) - –0.09983    

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