Math Problem Statement
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
Solution
To solve this problem, we need to find the ratio where
- (sum of reciprocals of cubes of all positive integers)
- (sum of reciprocals of cubes of all positive odd integers)
Step-by-Step Solution:
-
Understanding :
is the Riemann zeta function at , which is known as :
-
Understanding :
is the sum of reciprocals of cubes of all positive odd integers:
-
Expressing and :
To find a relationship between and , consider separating the terms of into sums over odd and even integers: Let's denote the sum over odd integers as , and the sum over even integers separately: Therefore,
-
Finding :
Using and the relation : Thus,
-
Finding :
Since , and . Therefore, .
Conclusion
The correct answer is:
(3) 15
Would you like further details on any step? Here are five related questions you might find interesting:
- What is the Riemann zeta function, and why is it significant in mathematics?
- Can you explain more about the convergence of series like ?
- How do properties of odd and even numbers affect summation in series?
- Are there similar problems that involve other values of for the Riemann zeta function?
- 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.
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Math Problem Analysis
Mathematical Concepts
Infinite series
Riemann zeta function
Formulas
\( \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} \)
Theorems
-
Suitable Grade Level
Advanced Mathematics
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