explain how to solve this question---Question Stem: $\frac{1}{1 \times 3} + \frac{1}{3 \times 5} + \frac{1}{5 \times 7} + \frac{1}{7 \times 9} + \frac{1}{9 \times 11} + \frac{1}{11 \times 13}$ 的计算结果是 Other Relevant Text: 的计算结果是 (meaning "the calculation result is")

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Let's solve this telescoping series step by step. We have a sum of fractions where each term has the form one over the product of two consecutive odd numbers. The key insight is to use partial fraction decomposition to reveal the telescoping pattern. Now let's apply partial fraction decomposition. For any term of the form one over k times k plus two, we can write it as one half times the difference of one over k minus one over k plus two. Let's apply this to our first few terms to see the pattern emerge. Now let's decompose all six terms in our series. Each fraction follows the same pattern. Notice how each decomposition creates two fractions, and when we add all these terms together, the middle terms will cancel out, leaving only the first and last terms. Now comes the magic of telescoping series. When we add all the decomposed terms together, we can factor out one half and observe the cancellation pattern. The negative one third from the first term cancels with the positive one third from the second term. Similarly, all the intermediate fractions cancel out, leaving only one over one minus one over thirteen. Now let's complete the calculation. We have one half times the quantity one minus one thirteenth. Converting to a common denominator, this becomes one half times twelve thirteenths, which equals twelve twenty-sixths. Simplifying by dividing both numerator and denominator by two, we get our final answer: six thirteenths.