f(x)=(3x+1) . a] Find the Taylor series expansion associated with f centered at 2. b] What is the largest interval on which the Taylor series expansion associated with f converges back to the function. Justify your answer as clearly as you can.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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f(x)=(3x+1) .

a] Find the Taylor series expansion associated with f centered at 2.
b] What is the largest interval on which the Taylor series expansion associated with f converges back to the function. Justify your answer as clearly as you can.

**Hint:** Observe that 

\[ f(x) = \frac{1}{49} \left( 1 + \frac{3(x - 2)}{7} \right)^{-2} . \]

This expression suggests a transformation applied to a function. The fraction and terms inside the brackets represent a shifted and scaled version of a parent function, and the negative exponent indicates an inverse squared relationship.
Transcribed Image Text:**Hint:** Observe that \[ f(x) = \frac{1}{49} \left( 1 + \frac{3(x - 2)}{7} \right)^{-2} . \] This expression suggests a transformation applied to a function. The fraction and terms inside the brackets represent a shifted and scaled version of a parent function, and the negative exponent indicates an inverse squared relationship.
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