Question

Determine the convergence of the given series using the Ratio Test. If the Ratio Test is inconclusive, state so and determine convergence with another test. $$\sum_{n=0}^{\infty} \frac{5^{n}-3 n}{4^{n}}$$

Short answer

The series diverges by the Divergence Test.

Step-by-step solution

Write Down the Given Series

The series given is \( \sum_{n=0}^{\infty} \frac{5^n - 3n}{4^n} \). Our task is to determine if this series converges or diverges.

Simplify the General Term

The general term of the series is \( a_n = \frac{5^n - 3n}{4^n} \). Simplifying, we get \( a_n = \left(\frac{5}{4}\right)^n - \frac{3n}{4^n} \).

Apply the Ratio Test

To use the Ratio Test, we calculate the limit of \( \left|\frac{a_{n+1}}{a_n}\right| \) as \( n \rightarrow \infty \). Substitute \( a_n = \left(\frac{5}{4}\right)^n - \frac{3n}{4^n} \) and \( a_{n+1} = \left(\frac{5}{4}\right)^{n+1} - \frac{3(n+1)}{4^{n+1}} \).

Calculate the Ratio of Consecutive Terms

Calculate \[ \frac{a_{n+1}}{a_n} = \frac{\left(\frac{5}{4}\right)^{n+1} - \frac{3(n+1)}{4^{n+1}}}{\left(\frac{5}{4}\right)^n - \frac{3n}{4^n}}. \] Expanding the terms, \( \frac{a_{n+1}}{a_n} \approx \frac{5}{4} - \frac{3}{4^{n+1}} \cdot \left(n+1\right) \) as \( n \rightarrow \infty \).

Evaluate the Limit

Evaluate the limit: \[ \lim_{n \rightarrow \infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n \rightarrow \infty} \left(\frac{5}{4} - \frac{3(n+1)}{4^{n+1}} \right) = \frac{5}{4}. \] This implies the Ratio Test is inconclusive since the limit \( \frac{5}{4} > 1 \).

Use an Alternative Convergence Test

Since the Ratio Test is inconclusive, use the Divergence Test. Examine the general term \( a_n = \left(\frac{5}{4}\right)^n - \frac{3n}{4^n} \). As \( n \rightarrow \infty \), \( \left(\frac{5}{4}\right)^n \) dominates and grows without bound, so \( a_n \) does not approach zero.

Conclusion about Convergence

Since \( a_n \) does not approach zero, the series \( \sum_{n=0}^{\infty} \frac{5^n - 3n}{4^n} \) diverges by the Divergence Test.

Key concepts

Ratio Test

The Ratio Test is a powerful tool for determining the convergence of infinite series, particularly those involving factorials or exponential functions. To use this test, consider an infinite series \( \sum_{n=0}^{\infty} a_n \). The Ratio Test involves calculating the limit:

• \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \)

When the limit \( L\) is:

• Less than 1, the series converges absolutely.
• Greater than 1, the series diverges.
• Equal to 1, the test is inconclusive.

In the original exercise, when applying the Ratio Test to the series \( \sum_{n=0}^{\infty} \frac{5^n - 3n}{4^n} \), the limit was found to be \( \frac{5}{4} \), which is greater than 1. Hence, the Ratio Test indicated that the series could be divergent, but additional checks were needed since the test was inconclusive in establishing absolute convergence.

Divergence Test

The Divergence Test, also known as the nth-term test for divergence, is a simple method to determine if an infinite series diverges. The essence of this test is:

• If the limit of the general term \( a_n \), as \( n \) approaches infinity, is not zero, \( \lim_{n \to \infty} a_n eq 0 \), then the series \( \sum_{n=0}^{\infty} a_n \) diverges.

In the case of the series examined, \( a_n = \left(\frac{5}{4}\right)^n - \frac{3n}{4^n} \). As \( n \) approaches infinity, the term \( \left(\frac{5}{4}\right)^n \) dominates because it grows exponentially and does not approach zero. Therefore, \( a_n \) itself does not approach zero, implying that the series diverges based on the Divergence Test. This conclusion serves as a straightforward check after other tests like the Ratio Test offer inconclusive results.

infinite series

An infinite series is simply the sum of an infinite sequence of terms. Formally, it is represented as \( \sum_{n=0}^{\infty} a_n \), where \( a_n \) are the terms of the sequence. Understanding convergence is key:

• If the sequence of partial sums \( S_n = \sum_{k=0}^{n} a_k \) approaches a finite limit as \( n \to \infty \), the series converges.
• If the partial sums do not settle to a limit, the series diverges.

In mathematics, testing for convergence helps verify whether an infinite series aggregates to a specific value or not. Infinite series are used in many applications, from solving calculus problems to modelling scientific phenomena. The series examined in the original exercise was \( \sum_{n=0}^{\infty} \frac{5^n - 3n}{4^n} \), and it was shown to diverge based on the tests applied.

dominance of terms

The concept of dominance in terms within a series refers to which part of the expression grows faster as the variable approaches infinity. In mixed expressions like \( \frac{5^n - 3n}{4^n} \), analyzing dominance is critical:

• Compare terms \( \left(\frac{5}{4}\right)^n \) and \( \frac{3n}{4^n} \).
• As \( n \to \infty \), \( \left(\frac{5}{4}\right)^n \) grows exponentially faster than any linear term like \( \frac{3n}{4^n} \).

Due to this, \( \left(\frac{5}{4}\right)^n \) dominates the behavior of the entire term \( a_n \), causing it to not approach zero. Recognizing which component of a term dominates helps predict the convergence behavior of the series. Such understanding is particularly useful when more complex series make reliance on initial tests less reliable.