`sum_(n=1)^oo1/(9n^2+3n-2)`
Let's rewrite the n'th term of the sequence as,
`a_n=1/(9n^2+3n-2)`
`=1/(9n^2+6n-3n-2)`
`=1/(3n(3n+2)-1(3n+2))`
`=1/((3n+2)(3n-1))`
Now let's carry out partial fraction decomposition,
`1/((3n+2)(3n-1))=A/(3n+2)+B/(3n-1)`
Multiply the above equation by LCD,
`1=A(3n-1)+B(3n+2)`
`1=3An-A+3Bn+2B`
`1=(3A+3B)n-A+2B`
Equating the coefficients of the like terms,
`3A+3B=0` -----------------(1)
`-A+2B=1` ------------------(2)
From equation 1,
`3A=-3B`
`A=-B`
Substitute A in equation 2,
`-(-B)+2B=1`
`B+2B=1`
`3B=1`
`B=1/3`
`A=-1/3`
`a_n=(-1/3)/(3n+2)+(1/3)/(3n-1)`
`a_n=1/(3(3n-1))-1/(3(3n+2))`
Now we can write down the n'th partial sum of the series as:
`S_n=(1/(3(3-1))-1/(3(3+2)))+(1/(3(3*2-1))-1/(3(3*2+2)))+..........+(1/(3(3n-1))-1/(3(3n+2)))`
`S_n=(1/6-1/15)+(1/15-1/24)+.........+(1/(3(3n-1))-1/(3(3n+2)))`
`S_n=(1/6-1/(3(3n+2)))`
`sum_(n=1)^oo1/(9n^2+3n-2)=lim_(n->oo)S_n`
`=lim_(n->oo)(1/6-1/(3(3n+2)))`
`=1/6`
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