`sum_(n=1)^oo 1/(9n^2+3n-2)` Find the sum of the convergent series.

Expert Answers

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Let's rewrite the n'th term of the sequence as,





Now let's carry out partial fraction decomposition,


Multiply the above equation by LCD,




Equating the coefficients of the like terms,

`3A+3B=0`    -----------------(1)

`-A+2B=1`  ------------------(2)

From equation 1,



Substitute A in equation 2,








Now we can write down the n'th partial sum of the series as:







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