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\( \frac { 1 } { 1.2 } + \frac { 1 } { 2.3 } + \frac { 1 } { 3.4 } + \ldots + \frac { 1 } { 2011.2012 } \)

Ask by Carlson Cole. in Vietnam
Jan 25,2025

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Tutor-Verified Answer

Answer

Tổng của dãy là \(\frac{2011}{2012}\).

Solution

Biểu thức đã cho là tổng của dãy số dạng \(\frac{1}{n(n+1)}\) từ \(n = 1\) đến \(n = 2011\). Ta có thể biến đổi từng hạng tử trong tổng như sau: \[ \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1} \] Vậy tổng của dãy sẽ là: \[ \sum_{n=1}^{2011} \left( \frac{1}{n} - \frac{1}{n+1} \right) \] Khi tiến hành tính tổng, các hạng tử giữa sẽ triệt tiêu nhau, chỉ còn lại: \[ \left( \frac{1}{1} - \frac{1}{2} \right) + \left( \frac{1}{2} - \frac{1}{3} \right) + \ldots + \left( \frac{1}{2011} - \frac{1}{2012} \right) = 1 - \frac{1}{2012} \] Do đó, tổng của dãy là: \[ 1 - \frac{1}{2012} = \frac{2011}{2012} \] **Kết quả:** Tổng của dãy là \(\frac{2011}{2012}\).

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The Deep Dive

To evaluate the sum \( S = \frac{1}{1.2} + \frac{1}{2.3} + \frac{1}{3.4} + \ldots + \frac{1}{2011.2012} \), we start by recognizing the pattern in the general term: \[ \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1} \] This transformation allows us to rewrite the sum: \[ S = \left( \frac{1}{1} - \frac{1}{2} \right) + \left( \frac{1}{2} - \frac{1}{3} \right) + \left( \frac{1}{3} - \frac{1}{4} \right) + \cdots + \left( \frac{1}{2011} - \frac{1}{2012} \right) \] By observing the structure of the sum, we notice that it is a telescoping series, where many terms cancel out: \[ S = 1 - \frac{1}{2012} \] After cancellation, we are left with: \[ S = \frac{2012 - 1}{2012} = \frac{2011}{2012} \] Thus, the final result is: \[ \boxed{\frac{2011}{2012}} \]

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