Das Quotientenkriterium

Consider an infinite series. To understand whether this infinite series converges to a finite value or grows endlessly, one uses the Ratio Test. This test examines the limit of the absolute value of the ratio between consecutive terms, called L.

If L is less than one, the series converges, meaning its sum reaches a finite number. If L is greater than one or equals infinity, the series diverges, meaning the sum grows without bound.

When L equals one, the test gives no clear result, and other methods are needed.

For example, take a principal amount invested that grows each year at a 4% compound interest rate. This growth creates a geometric sequence of terms, where each new term is 1.04 times the previous one.

Because this value is greater than one, each term increases with each compounding period, and the total sum grows without bound.

The series diverges, showing that the accumulated sum increases without limit, and its infinite total cannot be calculated.