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The sum of the infinite series 1+(1+1/5)...

The sum of the infinite series `1+(1+1/5)(1/2)+(1+1/5+1/(5^2))(1/(2^2))+...`

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To solve the infinite series \( S = 1 + \left(1 + \frac{1}{5}\right) \left(\frac{1}{2}\right) + \left(1 + \frac{1}{5} + \frac{1}{5^2}\right) \left(\frac{1}{2^2}\right) + \ldots \), we can follow these steps: ### Step 1: Identify the pattern in the series The series can be rewritten as: \[ S = 1 + \left(1 + \frac{1}{5}\right) \left(\frac{1}{2}\right) + \left(1 + \frac{1}{5} + \frac{1}{5^2}\right) \left(\frac{1}{2^2}\right) + \ldots \] We notice that the terms in parentheses form a geometric series. ...
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Knowledge Check

  • What is the sum of the infinite geometric series 2+(-(1)/(2))+((1)/(8))+(-(1)/(32))+… ?

    A
    `1(3)/(8)`
    B
    `1(2)/(5)`
    C
    `1(1)/(2)`
    D
    `1(3)/(5)`

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