Worksheet: Transcendental Functions as Power Series

In this worksheet, we will practice using a power series representation for exponential, cosine, and sine series to approximate values of transcendental functions.

Q1:

Consider 𝑓 ( 𝑥 ) = ( 2 𝑥 ) l n .

Find a power series representation for 𝑓 ( 𝑥 ) .

  • A 𝑓 ( 𝑥 ) = ( 2 ) 1 𝑛 𝑥 2 l n 𝑛 = 0 𝑛
  • B 𝑓 ( 𝑥 ) = ( 2 ) + 𝑥 2 l n 𝑛 = 0 𝑛 + 1
  • C 𝑓 ( 𝑥 ) = ( 2 ) 𝑥 2 l n 𝑛 = 1 𝑛 + 1
  • D 𝑓 ( 𝑥 ) = ( 2 ) 𝑥 2 ( 𝑛 + 1 ) l n 𝑛 = 0 𝑛 + 1 𝑛 + 1
  • E 𝑓 ( 𝑥 ) = ( 2 ) + 𝑥 2 ( 𝑛 + 1 ) l n 𝑛 = 1 𝑛 + 1 𝑛 + 1

Find its interval of convergence.

  • A | 𝑥 | < 2
  • B | 𝑥 | > 0
  • C | 𝑥 | > 1
  • D | 𝑥 | > 2
  • E | 𝑥 | < 1

Q2:

Consider 𝑔 ( 𝑥 ) = 𝑒 𝑥 .

Find a power series representation for 𝑔 ( 𝑥 ) .

  • A 𝑔 ( 𝑥 ) = 𝑥 𝑛 𝑛 = 0 𝑛 + 1
  • B 𝑔 ( 𝑥 ) = 𝑥 𝑛 𝑛 = 1 𝑛
  • C 𝑔 ( 𝑥 ) = 𝑥 𝑛 + 1 𝑛 = 0 𝑛 + 1
  • D 𝑔 ( 𝑥 ) = 𝑥 𝑛 𝑛 = 0 𝑛
  • E 𝑔 ( 𝑥 ) = 𝑥 𝑛 𝑛 = 1 𝑛 + 1

Use the first three terms of this series to find an approximate value of 𝑒 0 . 4 to 2 decimal places.

Q3:

The function can be represented by the power series . Use the first two terms of this series to find an approximate value of to two decimal places.

Q4:

The function can be represented by the power series . Use the first two terms of this series to find an approximate value of to 2 decimal places.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.