Use the Principle of Mathematical Induction to show that the...

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{n(n+1)}=\frac{n}{n+1}
$$

Key Concepts

Mathematical Induction

Mathematical induction is a fundamental proof technique used to demonstrate that a property or statement holds for all natural numbers. It consists of two main parts: proving the statement true for an initial value (base case) and proving that if the statement holds for an arbitrary natural number, it must also hold for the next (inductive step). This method is widely used across mathematics to establish the validity of infinite sequences of assertions.

Base Case

The base case in induction is the initial step where the statement is verified for the first natural number (often n=1). This case forms the foundation of the induction proof, ensuring that the property in question holds at the starting point of the natural numbers.

Inductive Hypothesis

The inductive hypothesis involves assuming that the statement holds for some arbitrary natural number (say, n=k). This assumption is key to setting up the inductive step, where one attempts to prove that the truth of the statement for n=k implies it holds for n=k+1.

Inductive Step

The inductive step is the process of proving that if the statement holds for an arbitrary natural number (as per the inductive hypothesis), then it must also hold for the next natural number. This step is crucial as it builds a bridge from the base case onward, ensuring that the property applies to all natural numbers by the principle of induction.

Telescoping Series

A telescoping series is a type of series in which many terms cancel out between consecutive partial sums, simplifying the overall expression significantly. Recognizing and utilizing the telescoping property can transform complex summations into manageable forms, often leading to a direct computation of the sum's closed form.

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