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A series is given by 1/3^1+2/3^2+3/3^3+4/3^4+5/3^5+....

Francine correctly applies the ratio test to determine whether the series converges or diverges.

 

Which statement reflects Francine's conclusion?

From the ratio test, r = 3. The series converges.

From the ratio test, r=1/3. The series diverges.

From the ratio test, r = 3. The series diverges.

From the ratio test, r=1/3. The series converges.

 May 15, 2020
 #1
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The nth term is n / 3n.

The (n + 1)th term is (n + 1) / 3n + 1.

 

The ratio test:  lim |   [ (n + 1) / 3n + 1 ]  /  [  n / 3n ]   |

                 =    lim |    [ (n + 1) / n ]  ·  [ 3n  / 3n + 1 ]   |

                 =    lim |    [ (n + 1) / n ]  ·  [  1/3  ]   |

                 =   (1/3) · lim |  (n + 1) / n  |

                 =   (1/3) · lim |  ( n/n ) + (1/n)  |

                 =   (1/3) · lim |  1 + (1/n)  |

                 =   (1/3) · [ lim | 1 | + lim | 1/n |

                 =   (1/3) · [ 1 + 0 |

                 =   (1/3)               <---  converges

 May 15, 2020

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