Question

Evaluate the limit of the following sequences. $$a_{n}=\frac{7^{n}}{n^{7} 5^{n}}$$

Short answer

Answer: The limit of the given sequence as \(n\) approaches infinity is 0.

Step-by-step solution

Simplify the sequence

First, let's simplify the expression by combining the exponential terms: $$a_{n} = \frac{7^{n}}{n^{7} 5^{n}} = \frac{(7/5)^{n}}{n^{7}}$$

Apply properties of limits

Using properties of limits and the limit of exponential functions, lets evaluate the limit as \(n\) approaches infinity: $$\lim_{n\to\infty} a_{n} = \lim_{n\to\infty} \frac{(7/5)^{n}}{n^{7}}$$

Use the limit comparison test

To analyze the limit, we can use the limit comparison test, where we compare the given limit with another limit. We will compare it with the geometric sequence limit: $$b_{n} = (7/5)^{n}$$ Since the limit of \(b_n\) is infinity (because the ratio of the geometric sequence is greater than 1), we can now evaluate: $$\lim_{n\to\infty} \frac{a_{n}}{b_{n}} = \lim_{n\to\infty} \frac{\frac{(7/5)^{n}}{n^{7}}}{(7/5)^{n}}$$

Simplify the ratio

Now, simplify the ratio by canceling out the common terms: $$\lim_{n\to\infty} \frac{\frac{(7/5)^{n}}{n^{7}}}{(7/5)^{n}} = \lim_{n\to\infty} \frac{1}{n^{7}}$$

Evaluate the limit

Finally, evaluate the limit: $$\lim_{n\to\infty} \frac{1}{n^{7}} = 0$$ Since the limit of the ratio is 0, the original limit \(\lim_{n\to\infty} a_{n}\) converges, and the limit is: $$\lim_{n\to\infty} \frac{7^{n}}{n^{7} 5^{n}} = 0$$

Key concepts

Exponential Growth

Exponential growth refers to the increase in quantity according to a constant multiplicative rate. Imagine it as your quantity speeding up very quickly – much like the snowball effect. In the context of sequences: when you see a term like \(7^n\), it indicates exponential growth because the base, 7, is consistently multiplied by itself. As \(n\) increases, the value of \(7^n\) rises steeply.
One key idea here is comparing exponential terms: in the provided sequence, you have terms like \(7^n\) and \(5^n\). You can simplify by combining them into one base, leading to \((7/5)^n\).
This simplification is useful because \((7/5)^n\) also represents exponential growth but at a rate influenced by the ratio of 7 to 5. Here, since 7 is greater than 5, \((7/5)^n\) grows exponentially.

• This behavior helps us analyze how quickly the sequence approaches infinity or some limit.
• It also helps decide the dominant term in a sequence, which can guide us in evaluating limits.

Limit Comparison Test

The limit comparison test is handy when comparing the behavior of two sequences as they approach infinity. It helps determine if both sequences converge or diverge based on their ratio. In simpler terms, you use a 'known' sequence to assess the unknown one.
In our sequence's solution, after simplifying the original sequence \(a_n\), the limit comparison test provides insight by contrasting \(a_n\) with a familiar sequence like \((7/5)^n\), known to diverge. If dividing one by the other yields a limit (0 in this case due to the polynomial \(n^7\) in the denominator), you can infer properties about \(a_n\) without direct calculation.

• Think of it as borrowing understanding from an easy sequence to help solve a tough one.
• Helps in deciding convergence by leaning on sequences whose limits are well-studied.

Geometric Sequences

Geometric sequences have a constant ratio between successive terms. Each term is the previous term multiplied by this constant, called the common ratio. If this ratio is greater than 1, the sequence grows, and if between -1 and 1, it converges to zero as terms progress.
For instance, the sequence \((7/5)^n\) is geometric. Here the common ratio is 7/5, greater than 1, making it an example of a diverging geometric sequence.
Knowing this, when you see such sequences in problems, pay attention to the ratio:

• It can tell you how the sequence behaves over time, whether shrinking or growing.
• The knowledge helps in easily comparing one sequence with another using the limit comparison test.

The link to exponential growth also comes in handy. Both concepts interpret a similar pattern that enhances our comprehension of sequences.