Question

Evaluate the limit of the following sequences or state that the limit does not exist. $$a_{n}=\frac{7^{n}}{n^{7} 5^{n}}$$

Short answer

Answer: The limit of the sequence as n goes to infinity is infinity (∞). The sequence does not converge to a finite value.

Step-by-step solution

Identify the type of limit

The given sequence is a limit of a sequence as n approaches infinity. We can write it as: $$\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{7^n}{n^7 5^n}$$

Use properties of limits

Use the properties of limits to simplify the expression: $$\lim_{n \to \infty} \frac{7^n}{n^7 5^n} = \lim_{n \to \infty} \frac{7^n}{5^n \cdot n^7} = \lim_{n \to \infty} \frac{\left(\frac{7}{5}\right)^n}{n^7}$$

Compare the growth rates of numerator and denominator

Observe the growth rates of the numerator and the denominator. The numerator has an exponential growth rate while the denominator has a polynomial growth rate. In general, exponential functions grow faster than polynomial functions when the base of the exponential function is greater than 1 (which is true for our case, as the base is \(\frac{7}{5}\)).

Evaluate the limit

As mentioned, the exponential function grows faster than the polynomial function in the denominator. As n approaches infinity, the exponential function's growth will dominate the polynomial function's growth. Therefore, the limit converges to: $$\lim_{n \to \infty} \frac{\left(\frac{7}{5}\right)^n}{n^7} = \infty$$ The limit of the sequence is infinity; it does not converge to a finite value.

Key concepts

Properties of Limits

Understanding the properties of limits is essential for evaluating the behavior of sequences as their terms grow larger and larger. One crucial aspect is that if the sequence \( a_n \) tends to infinity and we multiply it by a positive constant, the result will still be infinite. Additionally, if two sequences \( a_n \) and \( b_n \) both approach infinity, their sum and product also tend to infinity.

Another property is that for a nonzero constant \( c \), the limit of \( c \cdot a_n \) is the product of \( c \) and the limit of \( a_n \). These properties allow us to manipulate limits algebraically to simplify the evaluation process. Also, when dealing with sequences that include both polynomial and exponential components, as in the exercise, we rely on general knowledge that the exponential growth rate typically surpasses polynomial growth as \( n \) becomes large.

Exponential Growth

Exponential growth is characterized by the rate of change increasing over time — the higher the base value, the faster the growth. In the context of sequences, we consider growth of the form \( a_n = b^n \) where \( n \) is a natural number and \( b \) is the base. If \( b > 1 \), the sequence grows without bound as \( n \) approaches infinity.

In our exercise, \( b \) is represented by \( \frac{7}{5} \) which is greater than 1. This implies that \( 7^n \) grows much faster than any polynomial function of \( n \) in the long run. This fundamental aspect of exponential growth is what leads to the sequence's limit being infinite.

Polynomial Growth Rate

Polynomial growth refers to increases represented by a polynomial expression, where the highest power of the variable dictates the overall growth of the function. In simpler terms, a sequence with polynomial growth such as \( a_n = n^k \) with \( k \) being a positive integer will increase at a rate proportional to \( n^k \).

Although polynomial growth can be rapid, especially for higher powers, it is outpaced by exponential growth at some point. For instance, in the given exercise, \( n^7 \) grows quickly, but not as quickly as \( (\frac{7}{5})^n \) for sufficiently large \( n \), which explains why the infinite behavior dominates in the limit.

Evaluating Limits

Evaluating limits often involves a combination of intuition and algebraic manipulation. For sequences, we consider what happens as \( n \) becomes very large, or approaches infinity. Through this process, we determine if the sequence approaches a finite value (converges) or continues to increase without bound (diverges).

In the exercise, we use the properties of limits to combine exponential and polynomial terms, then observe their growth rates to conclude that the limit diverges to infinity. This evaluative step is critical for understanding the long-term behavior of the sequence and determining its overall trend.

Sequences and Series Calculus

Sequences and series are fundamental concepts in calculus, with a sequence being an ordered list of numbers and a series defined as the sum of a sequence's terms. Calculus provides tools to analyze these mathematical structures, particularly regarding their limits.

When working with sequences and series, understanding convergence and divergence is central. A series can converge to a specific sum, while a divergent series fails to settle on a finite total. In relation to our exercise, while we dealt with a single sequence, these concepts transfer to series, allowing us to understand more complex scenarios of summation and progression in calculus.