Answered: Taylor seriesa. Use the definition of a…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.3: Geometric Sequences

Problem 44E

See similar textbooks

Related questions

Question

Taylor series
a. Use the definition of a Taylor series to find the first four nonzero
terms of the Taylor series for the given function centered at a.
b. Write the power series using summation notation.

ƒ(x) = ln x, a = 3

Expert Solution

Step 1

Given:

$f (x) = \ln (x)$ and $a = 3$

We have to find:

(a) Use the definition of a Taylor series to find the first four nonzero
terms of the Taylor series for the given function centred at a.

(b) Write the power series using summation notation.

Step 2

(a) Now write the Taylor's Series formula with centre $a$ :

$f (x) = f (a) + \frac{f' (a)}{1!} (x - a) + \frac{f'' (a)}{2!} {(x - a)}^{2} + \frac{f''' (a)}{3!} {(x - a)}^{3} + . . . . . . . . .$

Here, $f (x) = \ln (x)$ and $a = 3$

Now,

$f (x) = f (a) = f (3) = \ln (3) f' (x) = \frac{1}{x} = f' (a) = f' (3) = \frac{1}{3} f'' (x) = \frac{- 1}{x^{2}} = f'' (a) = f'' (3) = \frac{- 1}{9} f''' (x) = \frac{2}{x^{3}} = f''' (a) = f''' (3) = \frac{2}{27}$

Hence, we get

$f (x) = \ln (3) + \frac{\frac{1}{3}}{1!} (x - 3) + \frac{\frac{- 1}{9}}{2!} {(x - 3)}^{2} + \frac{\frac{2}{27}}{3!} {(x - 3)}^{3} + . . . . . . . = \ln (3) + \frac{1}{3} (x - 3) - \frac{1}{18} {(x - 3)}^{2} + \frac{1}{81} {(x - 3)}^{3} + . . . . . .$

Hence, the Taylor's Series with first non-zero terms with centre $a = 3$ is

$f (x) = \ln (3) + \frac{1}{3} (x - 3) - \frac{1}{18} {(x - 3)}^{2} + \frac{1}{81} {(x - 3)}^{3} + . . . . . .$

Step by step

Solved in 3 steps

Knowledge Booster

$Power Series$

Learn more about

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

Recommended textbooks for you

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

SEE MORE TEXTBOOKS

Taylor seriesa. Use the definition of a Taylor series to find the first four nonzeroterms of the Taylor series for the given function centered at a.b. Write the power series using summation notation. ƒ(x) = ln x, a = 3