Express the following product as a single fraction: ( 1 + 1 3 ) ( 1 + 1 9 ) ( 1 + 1 81 ) ⋯ ( 1 + 1 3 ( 2 n ) )I'm having difficulty with this problem:What i did was:I rewrote the 1 as 3 3 here is what i rewrote the whole product as: ( 3 3 + 1 3 ) ( 3 3 + 1 3 2 ) ( 3 3 + 1 3 4 ) ⋯ ( 3 3 + 1 3 2 n ) but how would i proceed after this?

Question

Express the following product as a single fraction:  (  1  +      1    3    )  (  1  +      1    9    )  (  1  +      1    81    )  ⋯  (  1  +      1          3              (        2        n        )              )I&#039;m having difficulty with this problem:What i did was:I rewrote the   1 as       3    3  here is what i rewrote the whole product as:      (          3      3        +          1      3        )        (          3      3        +          1              3        2              )        (          3      3        +          1              3        4              )    ⋯      (          3      3        +          1              3                              2            n                                )  but how would i proceed after this?

mnaonavl · Accepted Answer

Recall the difference of squares formula      a    2    −      b    2    =  (  a  +  b  )  (  a  −  b  )  .With the special case   a  =  1,   b  =      x    n  , we would get  1  −      x          2      n        =  (  1  −      x    n    )  (  1  +      x    n    )  .What does this suggest?

aniawnua · Accepted Answer

Show that for       |    x      |    &amp;lt;  1 that      ∏          n      =      0        ∞    (  1  +      x                  2        n              )  =      ∑          n      =      0        ∞        x    n    .Then use that      ∑          n      =      0        ∞        x    n    =      1          1      −      x        .

Express the following product as a single fraction: ( 1 + 1 3 </mfrac>

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