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True or False?

For any positive real number r, {r^n} converges.

False: Take any positive r, then as n → ∞ r diverges.

If {x_n*y_n} converges, then {x_n} and {y_n} both converge.

False: Suppose {x_n} or {y_n} dont converge. Then take x_n = n^2 and y_n = 1/n. Then x_n*y_n = n, which diverges. Thus it does not converge.

Can anyone please verify this? Thank you.

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  • $\begingroup$ For the first, should not say take any positive $r$, for the sequence $(1/2)^n$ converges. $\endgroup$ – André Nicolas Oct 16 '14 at 23:18
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  1. this is definitely false: if $r = 2,r^n\to \infty$ and $(r^n)$ diverges. Another example is $r^n = (-1)^n$.

  2. this is false: consider $x_n = y_n = (-1)^n$. $x_ny_n \to 1$ but both $x_n$ and $y_n$ diverge.

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