By Alan M. Polansky

Sequences of actual Numbers and capabilities advent Sequences of actual Numbers Sequences of actual features The Taylor growth Asymptotic Expansions Inversion of Asymptotic ExpansionsRandom Variables and attribute services creation likelihood Measures and Random VariablesSome vital Inequalities a few restrict conception for EventsGenerating and attribute FunctionsConvergence of Random VariablesRead more...

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**Extra resources for Introduction to Statistical Limit Theory**

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Indeed, note that 1 −1/2 + cos(t) 2t = 1 + t1/2 cos(t), −1/2 t does not remain bounded as t → ∞. The reason that differentiation is not applicable here is due to the cyclic nature of G(t). As t → ∞ the t1/2 term dominates the sin(t) term in G(t), so this periodic pattern is damped out in the limit. However, the t−1/2 term, which converges to 0, in g(t) is dominated by the cos(t) term as t → ∞ so the periodic nature of the function results. 17 is not applicable in this case as g(t) is not strictly increasing.

For any value of n ∈ N, |fn (x) − 0| < ε on (0, 1) when xn < ε which implies that n > log(ε)/ log(x), where we note that log(x) < 0 when x ∈ (0, 1). Hence, such a bound on n will always depend on x since log(x) is unbounded in the interval (0, 1), and therefore the sequence of functions {fn }∞ n=1 does not converge uniformly to f as n → ∞. 12 demonstrates one characteristic of sequences of functions that are uniformly convergent, in that the limit of a sequence of uniformly convergent continuous functions must also be continuous.

3, and as such is an excellent example of using the limit infimum and supremum to compute a limit. 7. Define the sequence {xn }∞ ) . Then n=1 as xn = (1 + n lim xn = e. n→∞ Proof. 22. 22, note that when n ∈ N is fixed n (1 + n−1 )n = i=0 n −i n−i n (1) = i n i=0 n −i n . 8), SEQUENCES OF REAL NUMBERS 11 and note that i−1 n −i (n − i)! n−j 1 n = ≤ . (n − i)! j=0 n i! Therefore it follows that n (1 + n−1 )n ≤ i=0 1 ≤ e, i! for every n ∈ N. The second inequality comes from the fact that ∞ e= i=0 1 , i!