Use substitution to find the Taylor series at x = 0 of the function In (1+5x²). What is the general expression for the nth term in the Taylor series at x = 0 for In (1+5x²)? 00 ΣΠ n=1 What is the Taylor series for In (1+5x7) at x = 0? OA. -5x7 + O B. 5x7 OC. 5x7. - 52x14 2 52x14 5321 2 3 14 5x 2 OD. 1-5x + + 14 5x 53x21 3 2 5x27 3 + 5x21 3 54x28 4 5x 54x28 4 28 4 + ... +... + ...

Use the following information to complete parts a. and b. below. f(x) = 11 Inx, a = 5 a. Find the first four nonzero terms of the Taylor series for the given function centered at a. OA. The first four nonzero terms are B. The first four nonzero terms are ∞ Ο A. 11 In 5+ Σ (-1)* - k=0 8 11 OC. The first four nonzero terms are 11 In 5+ 5 D. The first four nonzero terms are 11 In 5- OB. 11 In 5+ (−1)k+1. k=1 11 5*(x-5)k 8 Ο c. 11In 5+ Σ k=1 11 5 11 11 5 b. Write the power series using summation notation. k.5k + -(x - 5)k 11(-1)+1(x-5)k 5k 11 50 (x-5)- 11 + (x-5)- 50 11 5 22 125 -(x-5)- 11 375 11 (x - 5)² + 50 (x - 5)² + ¯(x − 5)² + 33 125 11 2500 11 375 (x - 5)³. 125 ¯(x – 5)³. -(x - 5)³. (x - 5)-(x - 5)²-(x - 5)³.

Please answer both questions

Use the definition of a Taylor series to find the first four nonzero terms of the series for f(x) centered at the given value of a. (Enter your answers as a comma-separated list.) f(x) 3xe* a 0