How to apply the ratio test to determine if ∑ 1 ⋅ 3 ⋅ 5 ⋅ ⋅ ⋅ ( 2 n - 1 ) 1 ⋅ 4 ⋅ 7 ⋅ ⋅ ⋅ ( 3 n - 2 ) from n = [ 1 , ∞ ) is convergent to divergent?

Question

How to apply the ratio test to determine if       ∑                  1        ⋅        3        ⋅        5        ⋅        ⋅        ⋅                  (          2          n          -          1          )                            1        ⋅        4        ⋅        7        ⋅        ⋅        ⋅                  (          3          n          -          2          )                     from       n    =          [      1      ,      ∞      )       is convergent to divergent?

Destiney Manning · Accepted Answer

Suppose       S    =                  ∑                  r          =          1                ∞                    a      n            and                 L    =          lim              n        →        ∞                    |                        a                      n            +            1                                    a          n                    |      if L &amp;lt; 1 then the series converges absolutely; if L &amp;gt; 1 then the series is divergent; if L = 1 or the limit fails to exist the test is inconclusive.Series -      S    =                  ∑                  n          =          1                ∞                            1        ⋅        3        ⋅        5        ⋅        ...        ⋅                  (          2          n          -          1          )                            1        ⋅        4        ⋅        7        ⋅        ...        ⋅                  (          3          n          -          2          )                    Test limit      L    =          lim              n        →        ∞                    |                                                  1              ⋅              3              ⋅              5              ⋅              ...              ⋅                              (                2                n                -                1                )                            ⋅                              (                2                                  (                  n                  +                  1                  )                                -                1                )                                                    1              ⋅              4              ⋅              7              ⋅              ...              ⋅                              (                3                n                -                2                )                            ⋅                              (                3                                  (                  n                  +                  1                  )                                -                2                )                                                                                        1              ⋅              3              ⋅              5              ⋅              ...              ⋅                              (                2                n                -                1                )                                                    1              ⋅              4              ⋅              7              ⋅              ...              ⋅                              (                3                n                -                2                )                                                        |            L    =          lim              n        →        ∞                    |                        1          ⋅          3          ⋅          ...          ⋅                      (            2            n            -            1            )                                (            2                          (              n              +              1              )                        -            1            )                                    1          ⋅          4          ⋅          ...          ⋅                      (            3            n            -            2            )                                (            3                          (              n              +              1              )                        -            2            )                              ⋅                        1          ⋅          4          ⋅          7          ⋅          ...          ⋅                      (            3            n            -            2            )                                    1          ⋅          3          ⋅          5          ⋅          ...          ⋅                      (            2            n            -            1            )                              |            L    =          lim              n        →        ∞                    |                                                                  1                ⋅                3                ⋅                ...                ⋅                                  (                  2                  n                  -                  1                  )                                                                          (            2                          (              n              +              1              )                        -            1            )                                                                              1                ⋅                4                ⋅                ...                ⋅                                  (                  3                  n                  -                  2                  )                                                                          (            3                          (              n              +              1              )                        -            2            )                              ⋅                                                                  1                ⋅                4                ⋅                7                ⋅                ...                ⋅                                  (                  3                  n                  -                  2                  )                                                                                                                        1                ⋅                3                ⋅                5                ⋅                ...                ⋅                                  (                  2                  n                  -                  1                  )                                                                        |            L    =          lim              n        →        ∞                    |                        2                      (            n            +            1            )                    -          1                          3                      (            n            +            1            )                    -          2                    |                           =          lim              n        →        ∞                    |                        2          n          +          1                          3          n          +          1                    |                           =          lim              n        →        ∞                    |                        2          n          +          1                          3          n          +          1                    ⋅                                    1            n                                                1            n                              |                           =          lim              n        →        ∞                    |                        2          +                      1            n                                    3          +                      1            n                              |                           =          2      3

How to apply the ratio test to determine if sum (135* * * (2n-1))/(147* * * (3n-2)) from n=[1,oo) is convergent to divergent?

Answered question

Answer & Explanation

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