limn →∞((1)/(1 ·2)+(1)/(2 ·3)+(1)/(3 ·4)+…+(1)/(n(n+1))) is equal to (A) 1 (B) -1 (C) 0 (D) None

Step 1: We are given the series $frac{1}{1 cdot 2}+frac{1}{2 cdot 3}+frac{1}{3 cdot 4}+ldots+fra

Limn →∞((1)/(1 ·2)+(1)/(2 ·3)+(1)/(3 ·4)+…+(1)/(n(n+1))) is...

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Key Concepts

Telescoping Series

A telescoping series is one in which consecutive terms cancel each other out when the series is expanded. Recognizing that a series can telescope allows one to simplify the summation by only considering the initial and final terms of the partial sum, which significantly eases the calculation of its sum or its limit as the number of terms grows large.

Partial Sums and Series Convergence

The concept of partial sums involves summing the first n terms of a series and analyzing the behavior of these sums as n increases. For a convergent infinite series, the limit of the partial sums as n approaches infinity gives the sum of the series. This approach is fundamental in understanding the convergence or divergence of series in mathematical analysis.

Limits

The limit is a fundamental concept in calculus that describes the behavior or value of a function or sequence as its input or index approaches a particular point, often infinity. In the context of series, evaluating the limit of the sequence of partial sums determines whether the series converges to a specific value.

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