I need help finding out which of the statements in this problem is true given the sum of the Taylor series. Please show work/process so I can follow along and see what I did wrong. The following is the sum of the first four terms of a Taylor series of a function f ata = 1.1+3(x – 1) + 5(x – 1)² + (x – 1)³-Which one of the following statements must be true?(a) f"(1) = 5(b) f'(1) + f"(1) = 4(c) f(0) = 2(d) f"(1) + f"(1) = 16

Answered: The following is the sum of the first…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

I need help finding out which of the statements in this problem is true given the sum of the Taylor series. Please show work/process so I can follow along and see what I did wrong.

The following is the sum of the first four terms of a Taylor series of a function f at
a = 1.
1+3(x – 1) + 5(x – 1)² + (x – 1)³
-
Which one of the following statements must be true?
(a) f"(1) = 5
(b) f'(1) + f"(1) = 4
(c) f(0) = 2
(d) f"(1) + f"(1) = 16

Expert Solution

Step 1

Use formula for Taylor series:

$T (x) = \sum_{k = 0}^{n} \frac{f^{(k)} (a)}{k!} {(x - a)}^{k} = f (a) + f' (a) (x - a) + \frac{f'' (a)}{2!} {(x - a)}^{2} + \frac{f''' (a)}{3!} {(x - a)}^{3} + . . . + \frac{f^{(n)} (a)}{n!} {(x - a)}^{n}$

Step 2

Apply the series for n=4 and a=1:

$T (x) = \sum_{k = 0}^{4} \frac{f^{(k)} (1)}{k!} {(x - 1)}^{k} = f' (1) + f' (x - 1) + \frac{f'' (1)}{2!} {(x - 1)}^{2} + \frac{f''' (1)}{3!} {(x - 1)}^{3} + \frac{f^{''''} (a)}{4!} {(x - a)}^{4}$

Step by step

Solved in 3 steps

Knowledge Booster

$Series$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions