In Problems 47-50, suggest a formula for each expression,...

In Problems $47-50,$ suggest a formula for each expression, and prove your conjecture using mathematical induction, $n \in N$.
$$
\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{n(n+1)}
$$

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Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to establish that a given statement holds for all natural numbers. It involves two steps: the base case, where the statement is verified for the first natural number, and the inductive step, where one assumes the statement for an arbitrary natural number n and then proves it for n+1. This method is widely used in proving formulas summing sequences or series as it rigorously demonstrates the property for an infinite set of elements.

Telescoping Series

A telescoping series is a series in which many terms cancel out with preceding or succeeding terms, leaving only a few terms that determine the final sum. This technique is particularly useful when evaluating series sums that can be decomposed into differences of successive terms, effectively reducing the complexity of the summation by canceling intermediate terms.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to break down a complicated rational expression into a sum of simpler fractions. This technique is particularly useful in series and integral calculations because it can simplify complex expressions into forms that are easier to sum or integrate, often revealing properties like telescoping behavior that simplify the evaluation of the sum.

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