Here 1= limT_oo ar, for Since by binomial theorem we have 2=(1+1)2=1+T+ T(T+1) +1>T Hence, if we choose N=1/ 8 or large, we have, for T>N T M This complete the solution 6 The concept of a limit extends directly to sequences of real vectors. Let bT beak×1 vector with real elements br,i=1,…,k.Ifbr→b2,i=1,…,k,then br -b, where b has elements bi, i= 1,...,k. An analogous extensions applies Definition Given g:R→R4(k,l∈N)andb∈Rk. The function g is continous at b if for any sequence{br} such that bT→b,g(br)→g(b) The following definition compares the behavior of a sequence br with the ehavior of a power of T, say T, where A is chosen so that and TA behave similarly Definition (i). The sequence br is at most of order TA, denoted br=O(T), if for some finite real number A>0, there existes a finite interger N such that for all T>M IT-M|<△ (ii). The sequence br) is of order smaller than TA, denoted br =o(Ta), if for every real number 8>0, there existes a finite interger N(O) such that for al T≥N(6),|T-b<6,ie.,TMbr→0 As we have defined these notations, br =O(T ) if iT-AbrI is eventually bounded, whereas bT o(T)if T-AbT -0. Obviously, if bT o(T), thenHere 1 = limT→∞ aT , for |aT − 1| =     2 T − (−1)T 2 T − 1     = 1 2 T . Since by binomial theorem we have 2 T = (1 + 1)T = 1 + T + T(T + 1) 2 · · · +1 > T. Hence, if we choose N = 1/δ or large, we have, for T > N, |aT − 1| = 1 2 T < 1 T < 1 N ≤ δ. This complete the solution. The concept of a limit extends directly to sequences of real vectors. Let bT be a k × 1 vector with real elements bTi , i = 1, ..., k. If bTi → bi , i = 1, ..., k, then bT → b, where b has elements bi , i = 1, ..., k. An analogous extensions applies to matrices. Definition: Given g : Rk → Rl (k, l ∈ N) and b ∈ R k . The function g is continous at b if for any sequence {bT} such that bT → b, g(bT) → g(b). The following definition compares the behavior of a sequence {bT } with the behavior of a power of T, say T λ , where λ is chosen so that {bT } and {T λ} behave similarly. Definition: (i). The sequence {bT } is at most of order T λ , denoted bT = O(T λ ), if for some finite real number 4 > 0, there existes a finite interger N such that for all T ≥ N, |T −λ bT | < 4. (ii). The sequence {bT } is of order smaller than T λ , denoted bT = o(T λ ), if for every real number δ > 0, there existes a finite interger N(δ) such that for all T ≥ N(δ), |T −λ bT | < δ, i.e., T −λ bT → 0. As we have defined these notations, bT = O(T λ ), if {T −λ bT } is eventually bounded, whereas bT = o(T λ ) if T −λ bT → 0. Obviously, if bT = o(T λ ), then 2