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Home > SparkNotes > Math Study Guides > Taylor Series >

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The Taylor Series

< Previous Section
The Remainder Term

Next Section >
Some Common Taylor Series

Problems

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

Solution for Problem 1 >>

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 -

+

-

+ ^...

Solution for Problem 2 >>

< Previous Section
The Remainder Term

Next Section >
Some Common Taylor Series

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