a. Use the Taylor series for sin(x) to find the Taylor series for sin(x²). What is the interval of convergence for the Taylor series for sin(x²)? Explain. b. Integrate the Taylor series for sin(x²) term by term to obtain a power series ex- pansion for sin(x²) dx. c. Use the result from part (b) to explain how to evaluate ¹ sin(x²) dx. Determine sin(x²) dx to 3 decimal the number of terms you will need to approximate places.
a. Use the Taylor series for sin(x) to find the Taylor series for sin(x²). What is the interval of convergence for the Taylor series for sin(x²)? Explain. b. Integrate the Taylor series for sin(x²) term by term to obtain a power series ex- pansion for sin(x²) dx. c. Use the result from part (b) to explain how to evaluate ¹ sin(x²) dx. Determine sin(x²) dx to 3 decimal the number of terms you will need to approximate places.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:3.
We can use power series to approximate definite integrals to which known techniques
of integration do not apply. We will illustrate this in this exercise with the definite
•1
integral ²* sin(x²) ds.
a. Use the Taylor series for sin(x) to find the Taylor series for sin(x²). What is the
interval of convergence for the Taylor series for sin(x²)? Explain.
b. Integrate the Taylor series for sin(x²) term by term to obtain a power series ex-
pansion for sin(x²) dx.
•1
c. Use the result from part (b) to explain how to evaluate sin(x²) dx. Determine
sin(x²) dx to 3 decimal
the number of terms you will need to approximate
places.
-1
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