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Math Help - convergence: proofs and counterexamples

  1. #1
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    convergence: proofs and counterexamples

    For each of the following, prove or give a counterexample:

    A) If (Sn) and (Tn) are divergent sequences, then (Sn + Tn) diverges
    B) If (Sn) and (Tn) are divergent sequences, then (SnTn) diverges
    C) If (Sn) and (Sn + Tn) are convergent sequences, then (Tn) converges
    D) If (Sn) and (SnTn) are convergent sequences, then (Tn) converges
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  2. #2
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    Quote Originally Posted by luckyc1423 View Post
    For each of the following, prove or give a counterexample:

    A) If (Sn) and (Tn) are divergent sequences, then (Sn + Tn) diverges
    False consider,
    s_n=1,0,1,0,1,0,...
    t_n=-1,0,-1,0,-1,0,...
    B) If (Sn) and (Tn) are divergent sequences, then (SnTn) diverges
    False consider,
    s_n=t_n=(-1)^n/n

    C) If (Sn) and (Sn + Tn) are convergent sequences, then (Tn) converges
    If {s_n} converges and {s_n+t_n} convergesn then by theorems we have that,
    {s_n+t_n}-{s_n}={t_n} converges.

    D) If (Sn) and (SnTn) are convergent sequences, then (Tn) converges
    No.

    If s_n does not converges to zero then this is true.

    But sometimes no.

    Let s_n=0 and t_n=n.
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