Use mathematical induction to prove that each of the given...

Use mathematical induction to prove that each of the given statements is true for every positive integer $n .$
$$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{n(n+1)}=\frac{n}{n+1}$$

Key Concepts

Mathematical Induction

Mathematical induction is a fundamental proof technique used to establish the truth of an infinite sequence of statements. It involves first proving a base case is true for an initial value, and then proving that if the statement holds for an arbitrary integer n, it must hold for n+1. This technique is widely used because it allows one to extend a property from a single instance to an entire infinite set of natural numbers.

Base Case

The base case in a proof by induction involves verifying that the statement or formula holds true when n is at its smallest value (often n = 1). This step is crucial because it serves as the foundation upon which the inductive argument builds, ensuring that the property is valid at the starting point of the induction.

Inductive Step

The inductive step is the part of an inductive proof where one assumes that the statement holds for some arbitrary positive integer n (the induction hypothesis) and then demonstrates that this assumption implies the statement is also true for n+1. This step is essential to propagate the truth of the statement from one integer to the next, thereby covering the entire set of natural numbers.

Telescoping Series

A telescoping series is a series where many terms cancel out when the series is written in an expanded form, leading to a simplified sum. Recognizing a telescoping series can greatly simplify the process of evaluating or proving the sum of the series, by reducing it to just a few terms. This concept is useful in understanding how complex series can often collapse to a simpler form.

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