If 1 ^ - 3 + 2 ^ - 3 + 3 ^ - 3 +...... upto∞ terms = p and 1 ^ - 3

+ 3 ^ - 3 + 5 ^ - 3 +... upto∞ terms = q , then p/q is  a/b (where a, b are positive co-prime integers), then

a + b is

(1) 32

(2) 16

(3) 15


(4) 27

To solve this problem, we need to find the ratio \( \frac{p}{q} \) where

- \( p = 1^{-3} + 2^{-3} + 3^{-3} + \cdots \) (sum of reciprocals of cubes of all positive integers)
- \( q = 1^{-3} + 3^{-3} + 5^{-3} + \cdots \) (sum of reciprocals of cubes of all positive odd integers)

### Step-by-Step Solution:

1. **Understanding \( p \):**

   \( p \) is the Riemann zeta function at \( s = 3 \), which is known as \( \zeta(3) \):
   \[
   p = \zeta(3) = \sum_{n=1}^{\infty} \frac{1}{n^3}
   \]

2. **Understanding \( q \):**

   \( q \) is the sum of reciprocals of cubes of all positive odd integers:
   \[
   q = 1^{-3} + 3^{-3} + 5^{-3} + \cdots = \sum_{n=0}^{\infty} \frac{1}{(2n+1)^3}
   \]

3. **Expressing \( p \) and \( q \):**

   To find a relationship between \( p \) and \( q \), consider separating the terms of \( p \) into sums over odd and even integers:
   \[
   p = \sum_{n=1}^{\infty} \frac{1}{n^3} = \sum_{\text{odd } n} \frac{1}{n^3} + \sum_{\text{even } n} \frac{1}{n^3}
   \]
   Let's denote the sum over odd integers as \( q \), and the sum over even integers separately:
   \[
   \sum_{\text{even } n} \frac{1}{n^3} = \sum_{m=1}^{\infty} \frac{1}{(2m)^3} = \sum_{m=1}^{\infty} \frac{1}{8m^3} = \frac{1}{8} \sum_{m=1}^{\infty} \frac{1}{m^3} = \frac{\zeta(3)}{8}
   \]
   Therefore,
   \[
   p = q + \frac{\zeta(3)}{8}
   \]

4. **Finding \( \frac{p}{q} \):**

   Using \( p = \zeta(3) \) and the relation \( p = q + \frac{\zeta(3)}{8} \):
   \[
   \zeta(3) = q + \frac{\zeta(3)}{8}
   \]
   \[
   q = \zeta(3) - \frac{\zeta(3)}{8} = \frac{8\zeta(3) - \zeta(3)}{8} = \frac{7\zeta(3)}{8}
   \]
   Thus,
   \[
   \frac{p}{q} = \frac{\zeta(3)}{\frac{7\zeta(3)}{8}} = \frac{\zeta(3) \times 8}{7\zeta(3)} = \frac{8}{7}
   \]

5. **Finding \( a + b \):**

   Since \( \frac{p}{q} = \frac{8}{7} \), \( a = 8 \) and \( b = 7 \). Therefore, \( a + b = 8 + 7 = 15 \).

### Conclusion

The correct answer is:

**(3) 15**

Would you like further details on any step? Here are five related questions you might find interesting:

1. What is the Riemann zeta function, and why is it significant in mathematics?
2. Can you explain more about the convergence of series like \( \sum_{n=1}^{\infty} \frac{1}{n^3} \)?
3. How do properties of odd and even numbers affect summation in series?
4. Are there similar problems that involve other values of \( n \) for the Riemann zeta function?
5. What are some applications of the Riemann zeta function in physics or number theory?

**Tip:** When dealing with infinite series, breaking them into smaller components often helps in finding relationships between them.

Explore the solution to the problem involving infinite series and the Riemann zeta function. Learn about the convergence of series, the properties of odd and even integers in summation, and how to find the ratio \( \frac{p}{q} \) in such problems. Suitable for advanced mathematics students.

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