Properties Of Sequences

and so lim„_rc n/2 = -f oe. Example 3.13 Use the results that

Ii—cm n n— oc ii and result (iv) of theorem 3.2 to find f/i + 3)(n2 — ij

Solution

Since

we can write

(« + 3)(/r-l) // -f 3 n2 - 1 lira ----—--- hm - Iim - - -few n—cc n~ ii—oo tl n—oc /j

Another useful application of this theorem concerns the present-value formula developed in section 3.3:

Theorem 3.2 provides a proof of the claim that, if r > 0. then PV, -»0 as I -v oo, Since the denominator is a delinitely divergent sequence (if r >0) and the numerator is a constant, then part (v) of theorem 3.2 establishes die result.

Since sequences are functions with their domains being the set of positive integers, one can define characteristics of monotonicity and boundedness in an analogous manner as was done for general functions in chapter 2. This is done formally below. (The property of boundedness was addressed informally earlier in this chapter.)

The following theorem is of obvious use in determining whether certain sequences are convergent.

Theorem 3.3 A monotonic sequence is convergent if and only if it is bounded.

Example 3.14

Use theorem 3.3 to show that the sequence a„ = I /2'\ n = 1, 2.3 is convergent.

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