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∑ -1/2 3^(1 - n) ((-1)^n - 3), n, 1, ∞ =21 / 8

Note that we can split this  as

2/1  +  2/9   +  2/81   +   .....+         and

1/3  +  1/27   +  1/243  + .... +

The  sum of the first series  =         2/  [ 1 - 1/9]  

The sum of the second series  =     (1/3)  / [ 1 - 1/9 ] 

So.....the sum  =    [ 2 +1/3]  / [ 1 - 1/9]   =   [ 7/3 ] / [ 8/ 9 ] =        63/ 24    =   21 / 8