Find a Geometric Power Series for the Function Centered at 0

question 4

Multiple Choice

Find a geometric power series for the function centered at 0, (i) by the technique shown in Examples 1 and 2 and (ii) by long division. $$f ( x ) = \frac { 10 } { 4 - x }$$

• A)
• \(\sum _ { n = 0 } ^ { \infty } \frac { 5 } { 2 } \left( \frac { x } { 4 } \right) ^ { n } , | x | < 4\)
• B)
• \(\sum _ { n = 0 } ^ { \infty } \frac { 1 } { 4 } \left( \frac { x } { 4 } \right) ^ { n } , | x | < 4\)
• C)
• \(\sum _ { n = 0 } ^ { \infty } 10 \left( - \frac { x } { 4 } \right) ^ { n } , | x | < 4\)
• D)
• \(\sum _ { n = 0 } ^ { \infty } \frac { 5 } { 2 } ( - 4 x ) ^ { n } , | x | < 4\)
• E)
• \(\sum _ { n = 0 } ^ { \infty } \frac { 5 } { 2 } ( - x ) ^ { n } , | x | < 1\)

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Definitions:

Reptiles

A class of cold-blooded, egg-laying vertebrates with scaly skin, which includes snakes, lizards, crocodiles, and turtles.

Plants

Multicellular organisms in the kingdom Plantae that typically produce their own food through photosynthesis and have cell walls made of cellulose.

Fish

Aquatic, gill-bearing organisms that lack limbs with digits, comprising a large and diverse group of vertebrates inhabiting both fresh and saltwater environments.

Birds

Warm-blooded, feathered vertebrates that lay eggs and have wings, belonging to the class Aves.