Question

Using the remainder estimate for the integral test, find a value of n that will ensure that the error in the approximation s ≠ˆ sn is less than 0.0000001.

1) n > 11
2) n > 0
3) n > 6
4) n > -18
5) n > 18

Answer 1

To find a value of n that will ensure that the error in the approximation s ≠ˆ sn is less than 0.0000001, we can use the remainder estimate for the integral test.

Step-by-step explanation:

To find a value of n that will ensure that the error in the approximation s ≠ˆ sn is less than 0.0000001, we can use the remainder estimate for the integral test. The remainder estimate states that the error in the approximation is less than or equal to the value of the first omitted term in the series. So we need to find a term in the series that is less than 0.0000001.

Let's denote the nth term in the series as an. Using the formula for the nth term of the series, we have:

an = 1/np

where p is a positive exponent. We want to find a value of n such that an is less than 0.0000001.

For example, let's take p = 2. Plugging in p = 2 into the formula for an, we have:

an = 1/n2

Setting an less than 0.0000001, we get:

1/n2 < 0.0000001

From this inequality, we can solve for n:

n2 > 1/0.0000001

n2 > 10,000,000

n > √10,000,000

n > 3162.2776

Therefore, we need n > 3162.2776 to ensure that the error in the approximation is less than 0.0000001. Rounding up to the nearest integer, we get n > 3163. So the correct answer is n > 3163.