If `|a| < 1 and |b| < 1,` then the sum of the series `a(a+b)+a^2(a^2+b^2)+a^3(a^3+b^3)+.....oo` is

Options

A. `(a)/(1-a)+(ab)/(1-ab)`
B. `(a^(2))/(1-a^(2))+(ab)/(1-ab)`
C. `(b)/(1-b)+(a)/(1-a)`
D. `(b^(2))/(1-b^(2))+(ab)/(1-ab)`

Answer

To find the sum of the series \( S = a(a + b) + a^2(a^2 + b^2) + a^3(a^3 + b^3) + \ldots \), we can break it down step by step.

### Step 1: Rewrite the Series
The series can be rewritten as:
\[ S = \sum_{n=1}^{\infty} a^n (a^n + b^n) \]
This can be separated into two parts:
\[ S = \sum_{n=1}^{\infty} a^{2n} + \sum_{n=1}^{\infty} a^n b^n \]

### Step 2: Identify Each Series
1. The first series \( \sum_{n=1}^{\infty} a^{2n} \) is a geometric series with the first term \( a^2 \) and common ratio \( a^2 \).
2. The second series \( \sum_{n=1}^{\infty} a^n b^n \) is also a geometric series with the first term \( ab \) and common ratio \( ab \).

### Step 3: Sum of the First Series
The sum of the first series can be calculated using the formula for the sum of an infinite geometric series:
\[ \text{Sum} = \frac{a}{1 - r} \]
where \( a \) is the first term and \( r \) is the common ratio. For the first series:
\[ \sum_{n=1}^{\infty} a^{2n} = \frac{a^2}{1 - a^2} \]

### Step 4: Sum of the Second Series
Similarly, for the second series:
\[ \sum_{n=1}^{\infty} a^n b^n = \frac{ab}{1 - ab} \]

### Step 5: Combine the Results
Now, we can combine the results from both series:
\[ S = \frac{a^2}{1 - a^2} + \frac{ab}{1 - ab} \]

### Final Step: Simplify the Expression
Thus, the final expression for the sum of the series is:
\[ S = \frac{a^2}{1 - a^2} + \frac{ab}{1 - ab} \]

### Conclusion
The sum of the series \( S = a(a + b) + a^2(a^2 + b^2) + a^3(a^3 + b^3) + \ldots \) is:
\[ S = \frac{a^2}{1 - a^2} + \frac{ab}{1 - ab} \]