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. $$\sum_{n=1}^{\infty} \frac{5^{n}+n^{4}}{7^{n}+n^{2}}$$

Short answer

The series converges by the Ratio Test.

Step-by-step solution

Define the terms of the series

For the series \( \sum_{n=1}^{\infty} a_n \), we identify \( a_n = \frac{5^n + n^4}{7^n + n^2} \). We will use these terms in the Ratio Test.

Compute the ratio of consecutive terms

The Ratio Test requires computing the limit of \( \left| \frac{a_{n+1}}{a_n} \right| \) as \( n \to \infty \). Calculate:\[ \frac{a_{n+1}}{a_n} = \frac{\frac{5^{n+1} + (n+1)^4}{7^{n+1} + (n+1)^2}}{\frac{5^n + n^4}{7^n + n^2}} = \frac{(5^{n+1} + (n+1)^4)(7^n + n^2)}{(7^{n+1} + (n+1)^2)(5^n + n^4)} \].

Simplify the ratio

Simplify the expression:\[ \left( \frac{5^{n+1} + (n+1)^4}{5^n + n^4} \right) \cdot \left( \frac{7^n + n^2}{7^{n+1} + (n+1)^2} \right) \].Focus on the leading terms in the numerators and denominators, i.e., \( 5^{n+1}/5^n = 5 \) and \( 7^n/7^{n+1} = 1/7 \).Thus, approximate:\[ \frac{5^{n+1}}{5^n} \cdot \frac{7^n}{7^{n+1}} = 5 \cdot \frac{1}{7} = \frac{5}{7} \].

Evaluate the limit of the ratio

Evaluate:\[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{5^{n+1} + (n+1)^4}{5^n + n^4} \cdot \frac{7^n + n^2}{7^{n+1} + (n+1)^2} \right| \approx \frac{5}{7} \].Since the limit is \( \frac{5}{7} < 1 \), the series converges according to the Ratio Test.

Key concepts

series convergence

Series convergence is a fundamental concept in mathematics, especially when dealing with infinite series. An infinite series is the sum of infinitely many terms in sequence, such as \( \sum_{n=1}^{\infty} a_n \). A series converges when the sequence of partial sums approaches a specific value as the number of terms grows. This means that regardless of how many terms you add, the total sum will settle at a finite number.

In this exercise, we aim to determine the series' convergence using specific convergence tests. If a series converges, the sum is finite. If it doesn't, the series diverges, implying the sum is infinite or does not settle to a single value.

The Ratio Test is a common convergence test, but when it is inconclusive, other tests such as the Root Test, Integral Test, or Comparison Test might be alternative solutions. The essence of testing convergence lies in understanding how individual series terms behave as they approach infinity and ensuring the entire series does not wildly grow out of control.

infinite series

An infinite series is essentially the sum of the terms in an infinite sequence. Mathematically, it is represented as \( \sum_{n=1}^{\infty} a_n \). Unlike finite sums, these series have no end and can be tricky to evaluate.

Understanding infinite series often revolves around determining whether the series converges or diverges. Some series may look complicated, but they converge to a neat finite number. Others may seem simple but diverge, never settling on a particular value.

Consider a basic geometric series like \( \sum_{n=0}^{\infty} x^n \). This series converges to \( \frac{1}{1-x} \) for \( |x| < 1 \), meaning it approaches a specific number as more terms are added. That's the beauty of convergence. However, if \( |x| \geq 1 \), the sum becomes unbound, indicating divergence. Infinite series could appear in various forms, but the key interest lies in understanding their sum behavior.

limit evaluation

Limit evaluation is a crucial part of understanding the behavior of a sequence or series, especially as the index grows towards infinity. It helps determine whether a series converges by checking the behavior of terms.

The Ratio Test, used here, relies heavily on limit evaluation. For the test, you compute \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \) to assess convergence. This limit tells us how each term compares to the next in the context of the bigger picture of series convergence.

With the given series, we evaluated the limit and found it to be \( \frac{5}{7} \). Since this ratio is less than 1, the series converges. This outcome was straightforward because the limit was clearly evaluated.

• If the limit less than 1, the series converges.
• If the limit equals 1, the test is inconclusive, requiring another method.
• If the limit is greater than 1, the series diverges.

These outcomes make the Ratio Test a powerful tool in understanding and evaluating series.