Welcome to our exploration of telescoping series of fractions. A telescoping series is a special type of infinite series where consecutive terms cancel out, much like the sections of a collapsible telescope. This cancellation leaves us with only the first and last terms, making the series surprisingly easy to evaluate. Let's look at a classic example: the sum from n equals 1 to infinity of 1 over n times n plus 1.
Before we can fully appreciate telescoping series, we need to understand partial fraction decomposition. This technique allows us to break down complex fractions into simpler components. Let's decompose our fraction 1 over n times n plus 1. We assume it equals A over n plus B over n plus 1. Multiplying both sides by n times n plus 1, we get 1 equals A times n plus 1 plus B times n. Expanding and collecting terms, we have 1 equals A plus B times n plus A. Comparing coefficients, A plus B equals zero and A equals 1. Solving this system gives us A equals 1 and B equals negative 1. Therefore, our decomposition is 1 over n times n plus 1 equals 1 over n minus 1 over n plus 1.