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Use the identity 1/(k(k + 1)) = 1/k – 1/(k + 1) and Ex- ercise 35 to compute E- 1/(k(k + 1)).

Use the identity 1/(k(k + 1)) = 1/k – 1/(k + 1) and Ex- ercise 35 to compute E- 1/(k(k + 1)).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(36) Use the identity 1∕(k(k + 1)) = 1∕k − 1∕(k + 1) and Exercise 35 to compute ∑n 1∕(k(k + 1)).

**Transcription for Educational Website** **Mathematics Exercise: Telescoping Sums** **Exercise 35:** Show that the sum \(\sum_{j=1}^{n} (a_j - a_{j-1}) = a_n - a_0\), where \(a_0, a_1, \ldots, a_n\) is a sequence of real numbers. This type of sum is called *telescoping*. **Exercise 36:** Use the identity \(1/(k(k+1)) = 1/k - 1/(k+1)\) and apply Exercise 35 to compute \(\sum_{k=1}^{n} 1/(k(k+1))\). **Discussion:** In Exercise 35, the concept of telescoping sums is explored. A telescoping sum is one where consecutive terms cancel each other out, leaving only the first and last elements in the sequence. The result is a simplified expression involving only the initial and final terms of the sequence. In Exercise 36, you are asked to apply this concept using a given identity. The identity simplifies the terms so that they telescope, allowing for easy calculation of the sum from \(k=1\) to \(n\) for the expression \(1/(k(k+1))\).
Transcribed Image Text:**Transcription for Educational Website** **Mathematics Exercise: Telescoping Sums** **Exercise 35:** Show that the sum \(\sum_{j=1}^{n} (a_j - a_{j-1}) = a_n - a_0\), where \(a_0, a_1, \ldots, a_n\) is a sequence of real numbers. This type of sum is called *telescoping*. **Exercise 36:** Use the identity \(1/(k(k+1)) = 1/k - 1/(k+1)\) and apply Exercise 35 to compute \(\sum_{k=1}^{n} 1/(k(k+1))\). **Discussion:** In Exercise 35, the concept of telescoping sums is explored. A telescoping sum is one where consecutive terms cancel each other out, leaving only the first and last elements in the sequence. The result is a simplified expression involving only the initial and final terms of the sequence. In Exercise 36, you are asked to apply this concept using a given identity. The identity simplifies the terms so that they telescope, allowing for easy calculation of the sum from \(k=1\) to \(n\) for the expression \(1/(k(k+1))\).
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