91Ó°ÊÓ

An infinite geometric series is a sum of the form \(\sum_{n=0}^{\-infty} ar^n\), where \(a\) represents the first term, and \(r\) is the common ratio. It's called 'geometric' because each term after the first is derived by multiplying the previous term by the common ratio, akin to a geometric progression.

When dealing with series, it is crucial to determine whether they converge to a finite number or diverge to infinity. Convergence occurs when the subsequent terms get infinitesimally small, adding an insignificant amount to the total sum as they progress towards infinity.

In the example at hand, the infinite series \(\sum_{n=1}^{\infty} e^{-n}\) can be seen as a geometric sequence with the common ratio less than one (\(\frac{1}{e} < 1\)). This supports the indication that the series may converge. However, always remember that identifying the series type is just the first step towards conclusively determining convergence.

To specifically determine the convergence of a geometric series, we employ the Test for Geometric Series. According to this criterion, a series \(\sum_{n=0}^{\infty} ar^n\) converges if the absolute value of the common ratio \(\mid r \mid\) is less than one. This is because when \(\mid r \mid < 1\), the terms \(ar^n\) decrease in magnitude as \(n\) increases and eventually approach zero.

Applying the test to \(\sum_{n=1}^{\infty} e^{-n}\), we see that the common ratio is \(\frac{1}{e}\), which is indeed less than one, confirming that the series converges. It's essential to grasp this test as a quick and reliable method. If at any point, the common ratio is greater than or equal to one, the series will diverge, meaning it will not sum up to a finite value.

Once we identify that a geometric series is convergent by the Test for Geometric Series, we can proceed to calculate its sum. The sum of a convergent geometric series is given by the formula \(S = \frac{a}{1-r}\), where \(a\) is the first term, and \(r\) is the common ratio. This formula emerges from leveraging the properties of geometric series, capturing the value it approaches as the number of terms becomes infinitely large.

For example, in the series \(\sum_{n=1}^{\infty} e^{-n}\), the first term \(a\) is \(1\) (since \(e^0 = 1\)), and the common ratio \(r\) is \(\frac{1}{e}\). Plugging these into the sum formula yields \(S = \frac{1}{1-\frac{1}{e}}\), which is the finite value that the series converges to. Understanding this concept is key to finding the definite value a series is tending towards, making it a vital tool in analyzing the behavior of infinite geometric series.