What is a potential limitation of using higher-order Taylor series for approximation?
A. Increased computational cost
B. Reduced accuracy
C. Increased risk of divergence
D. Limited convergence range
Answer (2)
The potential limitation of using higher-order Taylor series for approximation is that they often have a limited convergence range, meaning they may only be accurate within a specific interval around the point of expansion. This can lead to inaccuracies when evaluating the function outside this interval. Thus, the best choice is D. Limited convergence range.
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In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. Higher-order Taylor series include more terms and derivatives, providing a more precise, local approximation of the function. However, there are potential limitations to using higher-order Taylor series for approximation.
One potential limitation is (A) Increased computational cost. This is because as more terms are added to the series, more calculations are needed, involving potentially complex derivatives. The computational overhead can increase significantly, especially for functions that require many terms for an accurate approximation.
Another limitation is (D) Limited convergence range. Even with more terms, the approximation only holds effectively close to the point of expansion. Farther from this point, the series may converge poorly or not at all, reducing its utility for wider intervals.
Therefore, while higher-order terms can improve the approximation near the expansion point, the increased computational cost and limited convergence range are critical factors to consider when using the Taylor series.
In summary, if you are choosing the most suitable limitation from the given options, (A) Increased computational cost is a relevant concern because performing the calculations for higher-order terms can be resource-intensive.
