Question

Show that the following sequence

pₙ = 1/2ₙ converges linearly to p=0. How large must n be before ∣pₙ−p∣≤5×10⁻²?

Answer 1

The sequence pₙ = 1/2ₙ converges linearly to p=0. To find the value of n for which |pₙ - p| ≤ 5×10⁻², we solve the inequality 1/(2ₙ) ≤ 5×10⁻² and find that n must be at least 5.

Step-by-step explanation:

To show that the sequence pₙ = 1/2ₙ converges linearly to p=0, we need to show that |pₙ - p| approaches zero as n approaches infinity. Here's how:

1. Let's calculate the difference between pₙ and p: |pₙ - p| = |1/2ₙ - 0| = 1/(2ₙ).
2. We want to find the value of n for which |pₙ - p| ≤ 5×10⁻². So we can write the inequality: 1/(2ₙ) ≤ 5×10⁻².
3. To solve the inequality, we can take the reciprocal of both sides and get: 2ₙ ≥ 1/(5×10⁻²) = 1/0.05 = 20.
4. Now, let's solve for n: 2ₙ = 20 ⟹ n = log₂(20) = 4.322.
5. Since n represents the number of terms in the sequence, we round up to the nearest whole number to ensure that we have enough terms. Therefore, n must be at least 5 to satisfy the inequality.