Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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)).
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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