While in this class we mainly deal with sequences that are d

While in this class we mainly deal with sequences that are defined explicitly there are another class of sequences out their that are defined inductively. For instance: Let a_n = { 1 if n = 1 2 Squareroot a_n - 1 if n > 1 Write out the first 5 terms of this sequence

Solution

Here the sequence is given by a_n = 1 if n=1

= 2 (a_n-1) if n > 1

(a) So the first 5 terms are a₁ = 1, a₂ = 2 (a₁) = 2 (1) = 2, a₃ = 2 (a₂) = 2 (2) = 2 2

a₄ = 2 (a₃) = 2 (2 2)

a₅ = 2 (a₄) = 2 (2 (2 2))

(b) A sequence {a_n } is said to coverge if it approaches to some limit. In other words, {f_n(x)} is said ito be convergent if f_n(x)  tends to l as n tends to .

(c) Let F be a complete ordered field such as R. Let (a_n) be a sequence of numbers from F such that (a_n) is bounded and monotonic nondecreasing. Then there is lF such that anl as n.

Here as n, Lim  _n a_n =  Lim  _n 2 ^{1+1/2+1/4+1/8+..........} = 2 ^1/(1-1/2) = 2² = 4