Question

Determine whether the sequence converges or diverges. If convergent, give the limit of the sequence. $$\left\\{a_{n}\right\\}=\left\\{\frac{4^{n}}{5^{n}}\right\\}$$

Short answer

The sequence converges to 0.

Step-by-step solution

Define the Sequence

The given sequence is \( \{a_n\} = \left\{ \frac{4^{n}}{5^{n}} \right\} \). Let's simplify this expression.

Simplify the Expression

Simplify the expression \( \frac{4^n}{5^n} \) to \( \left( \frac{4}{5} \right)^n \). This shows that the sequence can be written as \( a_n = \left( \frac{4}{5} \right)^n \).

Analyze the Base of Exponent

Notice that the base of the exponent is \( \frac{4}{5} \), which is a fraction less than 1. As \( n \) becomes very large, \( \left( \frac{4}{5} \right)^n \) becomes very small because each term is a fraction of the previous one.

Determine the Convergence or Divergence

Since \( \left( \frac{4}{5} \right)^n \to 0 \) as \( n \to \infty \), the sequence converges.

State the Limit

The limit of the sequence is 0. Therefore, this means that the sequence is convergent, with a limit of \( 0 \).

Key concepts

Limit of a Sequence

When studying sequences, a fundamental concept is determining the limit. The limit of a sequence describes the value that the terms in the sequence approach as the sequence progresses towards infinity.
In mathematical terms, a sequence \ \( \{ a_n \} \ \) converges to a limit \ \( L \ \) if, for every positive number \ \( \epsilon \ \), there exists a corresponding positive integer \ \( N \ \) such that for all \ \( n > N \ \), the absolute difference \ \( |a_n - L| \ \) is less than \ \( \epsilon \ \).

• This means the terms get arbitrarily close to \ \( L \ \) as \ \( n \ \) becomes large.
• The limit, if it exists, provides a definitive target that the elements of a sequence will approach.

Understanding the limit helps in determining the behavior and "end behavior" of a sequence, capturing its long-term trend.

Exponential Sequences

Exponential sequences are sequences where each term is obtained by raising a base to the power of some term index, usually denoted by \ \( n \ \). For example, the given sequence \ \( \{a_n\} = \left( \frac{4}{5} \right)^n \ \) is an exponential sequence where the base is \ \( \frac{4}{5} \ \).
Unlike linear sequences, exponential sequences can grow or decay rapidly, depending on the base.

• If the base is greater than 1, the sequence grows as \ \( n \ \) increases.
• If the base is a fraction between 0 and 1, the sequence decays and approaches zero as \ \( n \ \) increases.

In the case of \ \( \left( \frac{4}{5} \right)^n \ \), because \ \( \frac{4}{5} \ \) is less than 1, the terms shrink, making the sequence converge to zero. Exponential sequences are significant in modelling real-world phenomena like population growth or radioactive decay due to their dynamic nature.

Convergent Sequences

Convergent sequences are sequences whose terms approach a specific value, known as the limit, as the sequence progresses toward infinity.
For a sequence like \ \( \{a_n\} = \left( \frac{4}{5} \right)^n \ \), we established that it converges because as \ \( n \ \) increases, the terms \ \( \left( \frac{4}{5} \right)^n \ \) approach 0.

• A sequence that fails to approach a specific limit is said to diverge.
• The concept of convergence is critical in calculus and mathematical analysis for understanding continuity, series, and integrals.

Convergence allows mathematicians to apply certain theorems and techniques, making sequences easier to analyze and understand. For this sequence, identifying that it converges aids in predicting its long-term behavior without needing to calculate every term.