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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:

The function s i n π‘₯ can be represented by the power series ∞ 𝑛 = 0 𝑛 2 𝑛 + 1 ο„š ( βˆ’ 1 ) ( 2 𝑛 + 1 ) ! π‘₯ . Use the first two terms of this series to find an approximate value of s i n 0 . 5 to 2 decimal places.

Q2:

Consider 𝑓 ( π‘₯ ) = ( 2 βˆ’ π‘₯ ) l n .

Find a power series representation for 𝑓 ( π‘₯ ) .

  • A 𝑓 ( π‘₯ ) = ( 2 ) βˆ’ ο„š ο€Ό 1 𝑛  ο€» π‘₯ 2  l n ∞ 𝑛 = 0 𝑛
  • B 𝑓 ( π‘₯ ) = ( 2 ) + ο„š ο€» π‘₯ 2  l n ∞ 𝑛 = 0 𝑛 + 1
  • C 𝑓 ( π‘₯ ) = ( 2 ) βˆ’ ο„š ο€» π‘₯ 2  l n ∞ 𝑛 = 1 𝑛 + 1
  • D 𝑓 ( π‘₯ ) = ( 2 ) βˆ’ ο„š π‘₯ 2 ( 𝑛 + 1 ) l n ∞ 𝑛 = 0 𝑛 + 1 𝑛 + 1
  • E 𝑓 ( π‘₯ ) = ( 2 ) + ο„š π‘₯ 2 ( 𝑛 + 1 ) l n ∞ 𝑛 = 1 𝑛 + 1 𝑛 + 1

Find its interval of convergence.

  • A | π‘₯ | < 2
  • B | π‘₯ | > 0
  • C | π‘₯ | > 1
  • D | π‘₯ | > 2
  • E | π‘₯ | < 1

Q3:

Consider 𝑔 ( π‘₯ ) = 𝑒 π‘₯ .

Find a power series representation for 𝑔 ( π‘₯ ) .

  • A 𝑔 ( π‘₯ ) = ο„š π‘₯ 𝑛 ! ∞ 𝑛 = 0 𝑛 + 1
  • B 𝑔 ( π‘₯ ) = ο„š π‘₯ 𝑛 ! ∞ 𝑛 = 1 𝑛
  • C 𝑔 ( π‘₯ ) = ο„š π‘₯ ( 𝑛 + 1 ) ! ∞ 𝑛 = 0 𝑛 + 1
  • D 𝑔 ( π‘₯ ) = ο„š π‘₯ 𝑛 ! ∞ 𝑛 = 0 𝑛
  • E 𝑔 ( π‘₯ ) = ο„š π‘₯ 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.

Q4:

The function c o s π‘₯ can be represented by the power series ∞ 𝑛 = 0 𝑛 2 𝑛 ο„š ( βˆ’ 1 ) ( 2 𝑛 ) ! π‘₯ . Use the first two terms of this series to find an approximate value of c o s 0 . 5 to two decimal places.

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