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Decide whether the given statement is true or false. If it is true, prove.

If it is false, give a counterexample.

(a). If lim (n→∞)An = 0, then Σ(n=1~∞) An converges.

(b). If Σ(n=1~∞) An and Σ(n=1~∞) Bn both diverge, then

Σ(n=1~∞) (An+Bn) also diverges.

(c). If { An } is a increasing sequence, then lim(n→∞) An= ∞

(d)Σ(n=1~∞) (1+1/n)^n diverges.

1 Answer

  • Favorite Answer

    (a) F

    Consider series Σ(n=1~∞) 1/n , then 1/n → 0 as n → ∞ ,

    but Σ(n=1~∞) 1/n divergers. (Because Harmonic Series)

    (b) F

    Consider two series Σ(n=1~∞) 1/n and Σ(n=1~∞) (n-n^2) / n^3 ,

    Clearly they are both divergers , but

    Σ(n=1~∞) (1/n + (n-n^2) / n^3 ) = Σ(n=1~∞) 1/n^2 converges.

    (Because p-series)

    (c) F

    Consider sequence {A_n} , defined by A_n+1 = √(3*A_n) and A_1 = 1

    Clearly {A_n} is a increasing sequence , BUT

    lim(n→∞) A_n = 3

    (d) T

    Since lim (n→∞) (1+1/n)^n = e ≠ 0 , so that Σ(n=1~∞) (1+1/n)^n diverges.

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