Vähennyssarjassa

Vähennyssarjassa, also known as a telescoping series or collapsing series, is a mathematical concept used in the study of infinite series. It is a type of series where the terms cancel out in a specific pattern, leading to a finite sum. This phenomenon occurs when the series is constructed such that each term is the difference between two consecutive terms of another sequence.

The general form of a vähennyssarjassa can be written as:

S = (a1 - a2) + (a2 - a3) + (a3 - a4) + ...

Here, a1, a2, a3, ... are the terms of the sequence. Notice that each term in the series is the difference between two consecutive terms of the sequence. When the series is summed, the intermediate terms cancel out, leaving only the first and last terms:

S = a1 - aN

where aN is the last term of the sequence. This cancellation is what gives the series its name, as it "telescopes" or "collapses" into a simple expression.

Vähennyssarjassa is particularly useful in calculus for evaluating certain types of improper integrals and sums. It is also a fundamental concept in the study of sequences and series, and is often used to illustrate the principles of convergence and divergence.