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.True or False?

Question

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.True or False?

Malcolm Flores · Accepted Answer

1. This is definitely false: if   r  =  2  ,      r    n    →  ∞ and   (      r    n    ) diverges. Another example is       r    n    =  (  −  1      )    n  .2. This is false: consider       x    n    =      y    n    =  (  −  1      )    n  .       x    n        y    n    →  1 but both       x    n   and       y    n   diverge.

For any positive real number r, r^n converges. False: Take any positive r, then as n->infty r diverges. If {x_ny_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_ny_n = n, which diverges. Thus it does not converge.

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