Question

Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess.

Answer 1

Step 1:

The sequence is .

Write out the terms of the sequence.

Terms of the sequence :

,

,

,

,

,

,

, .....

The sequence, .

Step 2:

The graph of the sequence is :

Observe the graph, the terms are decreasing and perhaps approaches .

Thus, the sequence converges to .

Step 3:

To confirm the result, first decide whether the sequence is convergent or divergent.

If the sequence is convergent, then find the value of the limit.

If a sequence is convergent, then there exist , where is a constant.

The sequence is .

As , then .

Evaluate the limits.

Thus, the sequence converges to .

Solution:

The sequence converges to .