Series approximation

Objective. Choose terms and control approximation accuracy.

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Learn: the key idea

Series approximation uses a truncated number of terms from an expansion to estimate a value, with accuracy improving as more terms are included and worsening as |x| grows closer to the boundary of validity. Example: using (1+x)^(1/2)≈1+0.5x−0.125x² with x=0.02 approximates √1.02 to several decimal places, whereas using only the first two terms gives a slightly less accurate result. Choosing how many terms to use depends on the required accuracy stated in the question.

Worked example

Use series approximation with the first three terms of (1+x)^(1/3) to estimate the cube root of 1.03, giving your answer to 4 decimal places.

1 $Write the expansion: (1+x)^(1/3)\approx1+(1/3)x-(1/9)x²$
2 $Substitute x=0.03: (1/3)(0.03)=0.01$
3 $Compute second term: (1/9)(0.03)²=(1/9)(0.0009)=0.0001$
4 $Combine: 1+0.01-0.0001=1.0099$

Answer: 1.0099

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Common mistake. Students include terms beyond what was requested or stop too early without checking the required accuracy, giving an answer that does not match the specified number of decimal places.

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