Consider the expression where the process continues indefinitely.

a. Show that this expression can be built in steps using the recurrence relation a₀ = 1, a_{n + 1} = ?1 + a_n, for n = 0, 1, 2, 3, ?. . . Explain why the value of the expression can be interpreted as provided the limit exists.

b. Evaluate the first five terms of the sequence {a_n}.?

c. Estimate the limit of the sequence. Compare your estimate with (1 + ?5)/2, a number known as the golden mean.

d. Assuming the limit exists, use the method of Example 5 to determine the limit exactly.

e. Repeat the preceding analysis for the expressionwhere p > 0. Make a table showing the approximate value of this expression for various values of p. Does the expression seem to have a limit for all positive values of p?

Transcribed Image Text:

V1 + V1 + V₁ + VI lim an n→∞ Vp + Vp + Vp + √p +