Consider the sequence given by an = − n / 2n+1 for n ≥ 0 Determine the limit

Question

Let L be the limit you found above. Find the minimal index N so that for n ≥ N, we have | an − L | < 0.008.

Answer 1

To find the limit consider huge values for n.

The larger n the more accurate it to approximate

a_n = -n/(2n+1) = -n/2n = -1/2

Therefore the limit L = -1/2

Notice that for all n, a_n > L.

Therefore

|a_n - L| = a_n - L

Therefore the constraint |a_n - L| < 0.008 is equivalent to

a_n - L < 0.008

Substituting actual values

-n/(2n+1) + 1/2 < 0.008

Multiply both sides by 2(2n+1)

-2n + 2n + 1 < 0.032n + 0.016

Solve for n

n > (1 - 0.016)/0.032 = 30.75

Therefore N = 31