Limits of Functions: Quiz

Problem 5

Let C(n) be a function that adds the first n powers of 1/2: C(0) = (1/2)0 = 1, C(1) = (1/2)0 + (1/2)1 = 1 + 1/2 = 3/2, C(2) = (1/2)0 + (1/2)1 + (1/2)2 = 1 + 1/2 + 1/4 = 7/4, and so on.

  1. Find a formula for C(n).
  2. What is the limit of C(n) as n gets large?

Answers, problem 5

  1. Let's assume an initial formula, C(n) = 1/2n. When n = 0, C(n) = 1 and when n = 1, C(n) = 1/2. But the trend above states that C(n) = 3/2 when n = 1. We can see that 3/2 = 2 - 1/2. Let's consider the case when n = 2. C(n) = 1/4, but the trend above states that C(n) = 7/4 when n = 2, and not 1/4. However, we can see that 2 - 1/4 = 7/4. We can see therefore that the actual formula bfor C(n) is 2– 1/2n
  2. As n gets larger, 1/2n will tend to 0, making the limit of C(n) as n gets large equal to 2.

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