91Ó°ÊÓ

In mathematics, the convergence of a series refers to the behavior of the sum of its terms as the number of terms goes to infinity. For a geometric series to converge, the series must meet specific criteria. In our example series, \( \sum_{n=0}^{\infty}\left(-\frac{1}{2}\right)^{n}(x-3)^{n} \), the convergence is contingent upon the common ratio, \( r \), being less than 1 in absolute value.

The common ratio is given by \( r = -\frac{1}{2}(x-3) \). For this series, convergence occurs when:

• \( |-\frac{1}{2}(x-3)| < 1 \)
• This simplifies to \( |x-3| < 2 \)
• Which further simplifies to \( 1 < x < 5 \)

If these conditions are met, the series will converge. When a series converges, it means that as you keep adding more terms, the sum will approach a specific finite value, giving useful insights into the behavior of the series within a particular interval of \( x \).

The sum of an infinite geometric series that converges can be succinctly expressed using a formula. If the series converges, its sum is determined by dividing the first term by one minus the common ratio. The formula is:

\[S = \frac{a}{1-r}\]

In the context of the given series, we have:

• The first term, \( a = 1 \)
• Common ratio, \( r = -\frac{1}{2}(x-3) \)

Substituting these values, the sum \( S(x) \) becomes:

\[S(x) = \frac{1}{1 + \frac{1}{2}(x-3)} = \frac{2}{x-1}\]

This expression is valid for values of \( x \) that satisfy the interval \( 1 < x < 5 \). Within this interval, the sum will provide a reliable representation of the entire series. It is fascinating to observe how an infinite series can converge to such a neat equation!

The interval of convergence for a series is the range of values for which the series will converge. For our geometric series to converge, the calculated conditions create the interval \( 1 < x < 5 \).

Here's how we derive this interval:

• The inequality \( |x-3| < 2 \) is solved to find \(-2 < x-3 < 2\).
• This simplifies to the interval \( 1 < x < 5 \).

Understanding the interval of convergence is key because it tells us where the series behaves nicely and approaches a calculable sum.

Outside this interval, the series does not converge, meaning it can grow larger without approaching a finite value. Thus, the interval of convergence defines the boundary within which our derived sum formula \( \frac{2}{x-1} \) is applicable and valid. Knowing this helps in interpreting solutions accurately and applying them to real-world scenarios where convergence is necessary for meaningful results.