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Find the sum of the following series: 1...

Find the sum of the following series: `1/2+1/(3^2)+1/(2^3)+1/(3^4)+1/(2^5)+1/(3^6)+oo`

Text Solution

Verified by Experts

The given series can be expressed as the sum of teo infinite geometric series, shown below.
`{1/2 +1/3^(2)+1/2^(3)+1/3^(4)+1/2^(5)+1/3^(6)+...oo}`
`={1/2+1/2^(3)+1/2^(5)+...oo}+{1/3^(2)+1/3^(4)+1/3^(6)+..oo}`
`{"an infinite GP with "a=1/2 and r=1/4}`
`{"an infinite GP with "a=1/9 and r=1/9}`
`=((1//2))/((1-1/4))+((1//9))/((1-1/9))=((1//2))/((3//4))+((1//9))/((8//9))`
`=(1/2xx4/3)+(1/9xx9/8)=(2/3+1/8)`
`=(16+3)/24=19/24`.
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