Sum the following series to n terms and to infinity :
(i) `(1)/(1.4.7) + (1)/(4.7.10) + (1)/(7.10.13) + ...` (ii) `sum_(r=1)^(n) r (r + 1) (r + 2) (r + 3)` (iii) `sum_(r=1)^(n) (1)/(4r^(2) - 1)`

Answer

Let's solve the given series step by step.

### (i) Sum the series:
\[ S_n = \frac{1}{1 \cdot 4 \cdot 7} + \frac{1}{4 \cdot 7 \cdot 10} + \frac{1}{7 \cdot 10 \cdot 13} + \ldots \]

1. **Identify the pattern in the denominators**:
The denominators are products of three terms that are in arithmetic progression (AP). The first term of each product is \(3n - 2\), the second is \(3n + 1\), and the third is \(3n + 4\).

2. **Express the nth term**:
The nth term can be expressed as:
\[ T_n = \frac{1}{(3n - 2)(3n + 1)(3n + 4)} \]

3. **Partial fraction decomposition**:
We can express \(T_n\) using partial fractions:
\[ T_n = \frac{A}{3n - 2} + \frac{B}{3n + 1} + \frac{C}{3n + 4} \]

Multiplying through by the denominator \((3n - 2)(3n + 1)(3n + 4)\) and solving for \(A\), \(B\), and \(C\) will give us the coefficients.

4. **Sum the series**:
After finding the coefficients, we can sum the series:
\[ S_n = \sum_{k=1}^{n} T_k \]
This will involve telescoping series, where most terms will cancel.

5. **Evaluate the limit as \(n \to \infty\)**:
To find the sum to infinity, we take the limit:
\[ S_{\infty} = \lim_{n \to \infty} S_n \]