Convergence Test for a Geometric Series

Identify which convergence test to use for a geometric series involving terms like $\frac{2^n}{5^n}$.

When determining the convergence of a geometric series, it is crucial to recognize its structure and apply the correct convergence test accordingly. A geometric series is defined by its constant ratio between successive terms, which significantly simplifies the convergence analysis compared to other series types. The central idea is to focus on this constant ratio, usually denoted by r. The convergence of a geometric series depends on the value of this ratio; specifically, a geometric series converges if the absolute value of the common ratio is less than one. Conversely, if the absolute value is equal to or greater than one, the series diverges.

In the given series with terms like $\frac{2^n}{5^n}$, it's beneficial to simplify the ratio between terms and express it in its simplest form. Once identified as a geometric series by rewriting it as $\left(\frac{2}{5}\right)^n$, it becomes evident that the series converges or diverges based on the absolute value of $\frac{2}{5}$, which is less than one, implying convergence.

Understanding geometric series' convergence is a fundamental aspect of calculus, often serving as a steppingstone to more complex series and tests. This knowledge is not only critical for solving problems involving power series but also the basis for applications in calculus where series convergence is essential. Recognizing and simplifying terms to identify the series type reduces complexity and directs you to the appropriate convergence test, a vital skill in tackling advanced mathematics problems.

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