Question

For each of the following sequences, if the divergence test applies, either state that \(\lim _{n \rightarrow \infty} a_{n}\) does not exist or find \(\lim _{n \rightarrow \infty} a_{n} .\) If the divergence test does not apply, state why. \(a_{n}=\frac{2^{n}+3^{n}}{10^{n / 2}}\)

Short answer

The sequence converges to 0.

Step-by-step solution

Identify the Terms of the Sequence

Given the sequence \(a_{n} = \frac{2^n + 3^n}{10^{n/2}}\). We need to analyze the behavior of this sequence as \(n\) goes to infinity.

Simplify the Sequence

We notice that within \(2^n + 3^n\), the term \(3^n\) will dominate over \(2^n\) as \(n\) increases because \(3 > 2\). Thus, as \(n\) becomes very large, \(2^n + 3^n\) is approximately \(3^n\). Similarly, the denominator remains \(10^{n/2}\). Thus, we can approximate the sequence as: \[ a_{n} \approx \frac{3^n}{10^{n/2}}.\]

Evaluate the Dominant Behavior

Rewriting the expression \(\frac{3^n}{10^{n/2}}\) as \(\left( \frac{3}{\sqrt{10}} \right)^n\) suggests considering the base \(\frac{3}{\sqrt{10}} \approx 0.9487\), which is less than 1.

Determine Convergence

Since \(\left( \frac{3}{\sqrt{10}} \right) < 1\), raising it to a power \(n\) that goes to infinity results in the sequence approaching zero: \[ \lim_{{n \to \infty}} \left( \frac{3}{\sqrt{10}} \right)^n = 0. \]

Conclusion

The limit of the sequence \(a_n\) as \(n\) approaches infinity is 0. Therefore, the divergence test shows that the sequence converges to a limit.

Key concepts

Sequences in Calculus

In calculus, a sequence is simply an ordered list of numbers that follow a specific pattern or rule. Each number in the sequence is called a term, and is typically denoted as \(a_n\), where \(n\) represents the position of the term in the sequence. Sequences are fundamental in calculus as they serve as a foundation for understanding more complex mathematical concepts like series and functions.

• A sequence can be finite, where it has a definite number of terms, or infinite, where the terms continue indefinitely.
• The behavior of sequences is often analyzed by looking at how the terms behave as \(n\) becomes very large (approaches infinity).
• One common type of sequence is the arithmetic sequence, where each term after the first is obtained by adding a constant to the previous term.
• Another common type is the geometric sequence, where each term is obtained by multiplying the previous term by a fixed, non-zero number called the ratio.

In the given problem, for the sequence \(a_n = \frac{2^n + 3^n}{10^{n/2}}\), it is important to identify patterns and terms to find out how the sequence behaves.

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as the term number \(n\) increases indefinitely. Mathematically, if \(a_n\) is a sequence, then the limit is written as \(\lim_{{n \to \infty}} a_n\).

• When a sequence has a limit, it means that as you progress through its terms, they get arbitrarily close to some real number \(L\).
• If \(a_n\) does not approach a single number as \(n\) becomes very large, the limit does not exist.
• In the exercise provided, we approximate the sequence \(a_n\) as \( \left( \frac{3}{\sqrt{10}} \right)^n\), which helps determine its limit.

The sequence's limit in this case is zero because \(\left( \frac{3}{\sqrt{10}} \right) < 1\). So, let's see what this means. As \(n\) grows, the fraction to the power of \(n\) gets smaller and smaller, nearing zero.

Convergence of Sequences

The convergence of a sequence refers to whether the sequence has a limit. If a sequence converges, it means as you advance through the terms, they approximate closer to a specific number.

• A sequence that converges settles down to a single value, often observed as \(n\) approaches infinity.
• If a sequence does not settle at a specific number, it is said to diverge.
• A well-defined limit is an indicator of convergence.

For the given sequence \(a_n = \frac{2^n + 3^n}{10^{n/2}}\):

• We evaluated it using an approximation \(a_n \approx \left( \frac{3}{\sqrt{10}} \right)^n\).
• The value of this base, \(\left( \frac{3}{\sqrt{10}} \right)\), is less than 1, which suggests convergence as \(n\) increases.
• Therefore, this sequence converges to zero, confirming that it's settling towards a specific, finite value.

Convergent sequences are essential in calculus as they help in understanding continuous growth, functions, and series, laying the groundwork for integral and differential equations.