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Find the sum sum(n=1)^(oo)(6^(n))/((3^(n...

Find the sum `sum_(n=1)^(oo)(6^(n))/((3^(n)-2^(n))(3^(n+1)-2^(n+1)))`

Text Solution

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`a_(n)=6^(n)/((3^(n)-2^(n))(3^(n+1)-2^(n+1)))`
`=(3^(n)(3^(n+1)-2^(n+1))-3^(n+1)(3^(n)-2^(n)))/((3^(n)-2^(n))(3^(n+1)-2^(n+1)))`
`=1/(1-(2/3)^(n))-1/(1-(2/3)^(n+1))`
`thereforea_1=1/(1-(2/3)^(1))-1/(1-(2/3)^(2))`
`a_(2)=1/(1-(2/3)^(2))-1/(1-(2/3)^(3))`
..........
........
`a_(n)=1/(1-(2/3)^(n))-1/(1-(2/3)^(n+1))`
Adding we get
`S_(n)=1/(1-2/3)-3/(3-(2/3)^(n+1))`
`thereforeS_(oo)=lim_(ntooo)S_(n)=3-1=2`
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CENGAGE PUBLICATION-PROGRESSION AND SERIES-SOLVED EXAMPLES 5.15
  1. Find the sum sum(n=1)^(oo)(6^(n))/((3^(n)-2^(n))(3^(n+1)-2^(n+1)))

    Text Solution

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