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Question
Find the sum of the following serie to infinity:
`2/5 + 3/5^2 +2/5^3 + 3/5^4 + ... ∞.`
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Solution
Given, `S_∞ = 2/5 + 3/5^2 +2/5^3 + 3/5^4 + ...`
`S_∞ = (2/5 + 2/5^3 + ...∞) + (3/5^2 + 3/5^4 + ...∞)`
S∞ = S'∞ + S''∞
r' = `(2/5^3)/(2/5) = 1/5^2`
r'' = `(3/5^4)/(3/5^2) = 1/5^2`
`S_∞ = a/(1 - r) ...|r| < 1`
S∞ = `(2/5)/(1 - 1/5^2) + (3/5^2)/(1 - 1/5^2)`
S∞ = `(2/5)/(1 - 1/25) + (3/25)/(1 - 1/25)`
S∞ = `(2/5)/((25 - 1)/25) + (3/25)/((25 - 1)/25)`
S∞ = `(2/5)/(24/25) + (3/25)/(24/25)`
S∞ = `(2 × 25)/(5 × 24) + (3 × 25)/(25 × 24)`
S∞ = `(10)/(24) + (3)/(24)`
S∞ = `13/24`
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