Find the sum to infinity of the series ` 1 + (1+a)r+(1 + a + a ^(2))r^(2) + … ` where ` 0 lt a, r le 1 `

Explanation

To find the sum to infinity of the series \( S = 1 + (1 + a)r + (1 + a + a^2)r^2 + \ldots \), where \( 0 < a \) and \( r \leq 1 \), we will follow these steps:

### Step 1: Write the series in a more manageable form
The series can be expressed as:
\[ S = 1 + (1 + a)r + (1 + a + a^2)r^2 + (1 + a + a^2 + a^3)r^3 + \ldots \]
Notice that the coefficients of \( r^n \) are the sums of the first \( n \) terms of the sequence \( 1, a, a^2, a^3, \ldots \).
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