A telescoping series gets its name from the way terms cancel out, like a collapsing telescope. In these series, when we write out the terms, intermediate values disappear, leaving only the first and last terms. For example, the series one-half plus one-sixth plus one-twelfth can be rewritten as differences that telescope to give us an exact sum of 1.
Partial fraction decomposition is the key technique for creating telescoping series. We start with a fraction like one over n times n plus one, and decompose it into simpler fractions A over n plus B over n plus one. By cross multiplying and comparing coefficients, we find that A equals 1 and B equals negative 1, giving us the telescoping form.
Let's apply partial fraction decomposition to solve a concrete example. Consider the sum from n equals 1 to infinity of 1 over n times n plus 1. Using our decomposition, this becomes the sum of 1 over n minus 1 over n plus 1. When we write out the partial sums, we see the telescoping pattern clearly. The intermediate terms cancel, leaving us with 1 minus 1 over n plus 1, which approaches 1 as n goes to infinity.
Advanced telescoping patterns involve more complex fractions that require decomposition into three or more partial fractions. For example, 1 over n times n plus 1 times n plus 2 needs three terms. Using the cover-up method, we substitute strategic values to find the coefficients quickly. This gives us A equals one-half, B equals negative 1, and C equals one-half, creating a more complex but still telescoping pattern.
Telescoping series convergence depends on whether the remaining terms approach zero. They have many applications, from calculating Riemann sums to solving recurrence relations. To recognize telescoping potential, look for rational functions with factored denominators. Here's a complete example: the series with 2n plus 1 over n times n plus 1 times n plus 2 decomposes into telescoping terms, giving us an exact sum of 3. This demonstrates the power of telescoping series in finding precise values for infinite sums.