श्रेणी `(2)/(3)+(3)/(3^(2))+(2)/(3^(3))+(3)/(3^(4))+(2)/(3^(5))+(3)/(3^(6))+...oo` का अनन्त पदों तक योग ज्ञात कीजिए ।

Explanation

`(2)/(3)+(3)/(3^(2))+(2)/(3^(3))+(3)/(3^(4))+(2)/(3^(5))+(3)/(3^(6))+......oo.`
`=((2)/(3)+(2)/(3^(3))+(2)/(3^(5))+....oo)+((3)/(3^(2))+(3)/(3^(4))+(3)/(3^(6))+....oo)`
`=(2)/(3)(1+(1)/(3^(2))+(1)/(3^(4))+....oo)+(3)/(3^(2))(1+(1)/(3^(2))+(1)/(3^(4))+....oo)`
`=((2)/(3)+(1)/(3))(1+(1)/(3^(2))+(1)/(3^(4))+....oo)`
`=((2+1)/(3))[(1)/(1-((1)/(9)))]=1.((1)/((8)/(9)))=(9)/(8).`