Limit of Sequence Problems

Solution of exercise 1

Prove that the sequence has a limit of 2. Also, calculate the terms whose distance from 2 is less than 0.1.

From_{a41 the distance to 2 is less than 0.1.}

Solution of exercise 2

Prove that the sequence has a limit of 4 and calculate how many terms of the succession are not within (4 − 0.001, 4 + 0.001).

The first thousand terms of the sequence are out.

Solution of exercise 3

Prove that the sequence has a limit of 1 and calculate how many terms of the succession are not within (1 − 0.001, 1 + 0.001).

The first 54 terms are out.

Solution of exercise 4

Prove that . Also, calculate the terms whose distance from the limit is less than 0.01.

From a₂₁₉ the distance to the limit is less than 0.01.

Solution of exercise 5

Prove that the sequence has a limit of +∞ and determine how many terms in the sequence are less than a million?

The 1,999 first terms of the sequence.

Solution of exercise 6

Prove that the sequence a_n= −n² has a limit of −∞. Also, what term of the sequence produces values of less than −10,000?

−1, −4, −9, −16, −25, −36, −49, ...

If N = 10,000, its square root is 100, therefore, a₁₀₁ will be less than −10,000.

a₁₀₁= −101² = −10,201