Exploring the Sum of Series: 1/13 1/35 1/57 ...
The problem at hand involves evaluating the sum of the series:
S sum_{n1}^{100} frac{1}{2n-12n1}
Simplifying Each Term
To simplify each term in the series, we use the method of partial fractions. Let's start by expressing the general term.
The term frac{1}{2n-12n1} can be simplified as follows:
frac{1}{2n-12n1} frac{1}{2} left(frac{1}{2n-1} - frac{1}{2n1}right)
Expanding the Sum
Using the derived form, we can now rewrite the sum S:
S sum_{n1}^{100} frac{1}{2} left(frac{1}{2n-1} - frac{1}{2n1}right)
By factoring out the frac{1}{2}, we get:
S frac{1}{2} sum_{n1}^{100} left(frac{1}{2n-1} - frac{1}{2n1}right)
Telescoping Series
Expanding the sum, we observe a telescoping series:
S frac{1}{2} left(left(frac{1}{1} - frac{1}{3}right) left(frac{1}{3} - frac{1}{5}right) left(frac{1}{5} - frac{1}{7}right) ldots left(frac{1}{199} - frac{1}{201}right)right)
The terms cancel out, leaving:
S frac{1}{2} left(1 - frac{1}{201}right)
Final Calculation
Carrying out the final calculation, we get:
S frac{1}{2} left(frac{201 - 1}{201}right) frac{1}{2} left(frac{200}{201}right) frac{100}{201}
General Implications and Applications
The method used here, namely partial fractions and telescoping series, is applicable to a broad range of problems involving sums of sequences. The approach can be generalized to various similar problems in mathematics and can be particularly useful in discrete mathematics, calculus, and algorithm design.
Additional Insights
For a deeper understanding, consider similar problems where the series terms follow a specific pattern that allows for telescoping. This technique can be extended to more complex series where manual calculation might be challenging.
Computational Verification
Using a computer program, the sum can be computed to verify the result. The sum evaluates to approximately 0.49751243781094495, which aligns with our analytical solution.
Conclusion
In conclusion, understanding and applying the techniques of partial fractions and telescoping series is crucial for solving complex series problems. These methods not only provide a clear pathway to the solution but also enhance problem-solving skills in mathematics and computer science.
References:
For further reading, consult textbooks on discrete mathematics, calculus, and series summation techniques. Online resources and mathematical software tools can also provide valuable insights and practical applications.