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The sum to infinity of the series 1+(4...

The sum to infinity of the series
`1+(4)/(5)+(7)/(5^(2))+(10)/(5^(3))+ . . . ,` is

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(ii) ` therefore S_(infty)=1+4((1)/(5))+7((1)/(5))^(2)+10((1)/(5))^(3)+"...." "upto "infty ".....(iii)"`
Multiplying both sides of Eq. (i) by `(1)/(5)`, we get
`(1)/(5) S_(infty)=((1)/(5))+4((1)/(5))^(2)+7((1)/(5))^(3)+"...." "upto "infty ".....(iv)"`
Subtracting Eq. (iv) from Eq. (iii), we get
`(1-(1)/(5)) S_(infty)=1+3[((1)/(5))+((1)/(5))^(2)+((1)/(5))^(3)+"...." "upto "infty "]`
`=1+3(((1)/(5))/(1-(1)/(5)))=1+(3)/(4)`
`=(4)/(5)S_(infty)=(7)/(4)=`
`therefore S_(infty)=(35)/(16)`
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