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given-two-sequence-a-n-gt-0-b-n-gt-0-such-n-N-a-n-n-lt-b-n-lt-a-n-1-n-a-if-n-1-a-n-converge-then-n-1-b-n-converge-b-proof-that-if-a-n-0-1-then-b-n-0-1-c-




Question Number 676 by 123456 last updated on 22/Feb/15
given two sequence a_n >0,b_n >0  such  ∀n∈N^∗ ,a_n ^n <b_n <a_n ^(1/n)   a. if Σ_(n=1) ^(+∞) a_n  converge then Σ_(n=1) ^(+∞) b_n  converge?  b. proof that if a_n ∈(0,1) then b_n ∈(0,1)  c. (dis)proof that if a_n →1 then b_n →1
giventwosequencean>0,bn>0suchnN,ann<bn<an1/na.if+n=1anconvergethen+n=1bnconverge?b.proofthatifan(0,1)thenbn(0,1)c.(dis)proofthatifan1thenbn1
Commented by prakash jain last updated on 22/Feb/15
b. 0<a_n <1⇒0<a_n ^n <1       0<a_n <1⇒0<a_n ^(1/n) <1       a_n ^n <b_n <a_n ^(1/n) ⇒0<b_n <1  c. ∵ a_n →1⇒ a_n ^n →1 and a_n ^(1/n) →1        ∴ a_n →1⇒b_n →1  a. If a_n  converges we can say a_n ^n  converges.        but we cannot conclude anything about        a_n ^(1/n) . Hence we cannot conclude that b_n         onverges.
b.0<an<10<ann<10<an<10<an1/n<1ann<bn<an1/n0<bn<1c.an1ann1andan1/n1an1bn1a.Ifanconvergeswecansayannconverges.butwecannotconcludeanythingaboutan1/n.Hencewecannotconcludethatbnonverges.

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