For the sequence, determine if the divergence test applies and either state that [tex] \lim _{n \rightarrow \infty} a_n [/tex] does not exist or find [tex] \lim _{n \rightarrow \infty} a_n [/tex]. (If an answer does not exist, enter DNE.)
[tex]
\begin{array}{r}
a_n=\frac{2^n+3^n}{50^{n / 2}} \
\lim _{n \rightarrow \infty} a_n=\square
\end{array}
[/tex]
The divergence test applies.
The divergence test does not apply.
Answer (2)
Rewrite the sequence a n = 5 0 n /2 2 n + 3 n as a n = ( 5 2 2 ) n + ( 5 2 3 ) n .
Simplify the fractions to get a n = ( 5 2 ) n + ( 10 3 2 ) n .
Find the limit as n approaches infinity: lim n → ∞ a n = 0 + 0 = 0 .
State that the divergence test applies and the limit is 0 .
Explanation
Problem Setup We are given the sequence a n = f r a c 2 n + 3 n 5 0 n /2 and asked to find lim n → ∞ a n . We also need to determine if the divergence test applies.
Rewriting the Sequence First, let's rewrite the expression for a n to make it easier to analyze: a n = 5 0 n /2 2 n + 3 n = ( 5 0 1/2 ) n 2 n + 3 n = ( 50 ) n 2 n + 3 n = ( 5 2 ) n 2 n + 3 n Now, we can separate the fraction into two parts: a n = ( 5 2 ) n 2 n + ( 5 2 ) n 3 n = ( 5 2 2 ) n + ( 5 2 3 ) n We can simplify the fractions further: 5 2 2 = 5 ⋅ 2 2 2 = 5 2 ≈ 0.283 5 2 3 = 5 ⋅ 2 3 2 = 10 3 2 ≈ 0.424
Finding the Limit Now we can rewrite a n as: a n = ( 5 2 ) n + ( 10 3 2 ) n Since both 5 2 < 1 and 10 3 2 < 1 , we can find the limit as n approaches infinity: n → ∞ lim ( 5 2 ) n = 0 n → ∞ lim ( 10 3 2 ) n = 0
Applying the Divergence Test Therefore, the limit of the sequence a n as n approaches infinity is: n → ∞ lim a n = n → ∞ lim ( 5 2 ) n + n → ∞ lim ( 10 3 2 ) n = 0 + 0 = 0 Since the limit exists and is equal to 0, the divergence test applies. The divergence test states that if lim n → ∞ a n = 0 , then the series ∑ a n diverges. However, if lim n → ∞ a n = 0 , the test is inconclusive, and we cannot determine whether the series converges or diverges using this test alone.
Final Answer The limit of the sequence a n as n approaches infinity is 0. The divergence test applies.
Conclusion The limit of the sequence is 0 . The divergence test applies.
Examples
Consider a scenario where you are analyzing the long-term behavior of an investment portfolio. Suppose the terms of a sequence represent the yearly gains of your portfolio, and you want to know if these gains will diminish over time. By finding the limit of the sequence, you can determine if the gains converge to a specific value or approach zero. If the limit is zero, it suggests that the portfolio's growth is slowing down, which might prompt you to re-evaluate your investment strategy. This kind of analysis is crucial for making informed financial decisions and planning for the future.
The limit of the sequence a n as n approaches infinity is 0 , and the divergence test applies. Since both components of the sequence approach zero, we conclude that lim n → ∞ a n = 0 . Therefore, we state that the divergence test applies here.
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