Here are the essential concepts you must grasp in order to answer the question correctly.
Mathematical Induction
Mathematical induction is a proof technique used to establish the truth of an infinite sequence of statements. It involves two main steps: the base case, where the statement is verified for the initial value (usually n=1), and the inductive step, where one assumes the statement holds for n=k and then proves it for n=k+1. This method is particularly useful for proving statements about integers.
Recommended video:
Series and Summation
A series is the sum of the terms of a sequence, and summation notation is used to represent this compactly. In the given question, the series involves fractions of the form 1/(n(n+1)), which can be simplified using partial fraction decomposition. Understanding how to manipulate and sum series is crucial for evaluating the left-hand side of the equation.
Recommended video:
Geometries from Conic Sections
Partial Fraction Decomposition
Partial fraction decomposition is a technique used to break down complex rational expressions into simpler fractions that are easier to work with. For the series in the question, expressing 1/(n(n+1)) as A/n + B/(n+1) allows for easier summation of the series. This method is essential for simplifying the left-hand side of the equation to match the right-hand side.
Recommended video:
Decomposition of Functions