Suppose that an=
(
1
+
ln
⁢3
𝑛
)
3
𝑛
.

What is the limit of the sequence \{a_n\}_{n=1}^\infty ?

Type your answer in the box. It should be a whole number.

Question

Suppose that an=
(
1
+
ln
⁢3
𝑛
)
3
𝑛
.

What is the limit of the sequence \{a_n\}_{n=1}^\infty ?

Type your answer in the box. It should be a whole number.

BlackTom AI · Accepted Answer

The standard exponential limit (1 + x/n)^n → e^x as n → ∞ is the key fact.

1) Relevant knowledge:
- Limit: lim_{n→∞} (1 + x/n)^n = e^x for any fixed real x.
- Properties of exponentials and logarithms: e^{ln a} = a, and (A^n)^m = A^{nm}.

2) Step-by-step reasoning:
Start with a_n = (1 + (ln 3)/n)^{3n}. Observe that 3n = 3·n, so this can be rewritten as a power of (1 + (ln 3)/n)^n:
a_n = [(1 + (ln 3)/n)^n]^3.

3) Intermediate limit calculation:
Apply the known limit with x = ln 3:
lim_{n→∞} (1 + (ln 3)/n)^n = e^{ln 3} = 3.
Therefore lim_{n→∞} a_n = [lim_{n→∞} (1 + (ln 3)/n)^n]^3 = 3^3 = 27.

(Alternatively, one can use logarithms: ln a_n = 3n·ln(1 + (ln 3)/n). For small t, ln(1+t) ~ t, so ln a_n → 3n·(ln 3 / n) = 3 ln 3, giving a_n → e^{3 ln 3} = e^{ln 27} = 27.)

4) Why this is correct:
The factorization into [(1 + (ln 3)/n)^n]^3 isolates the canonical form (1+x/n)^n whose limit is e^x. Raising that limit to the third power yields the final value. The alternative logarithmic approach confirms the same result.

5) Common incorrect ideas:
- Mistaking the exponent so as to get e^{ln 3} = 3 (forgetting the extra factor 3) would give 3 instead of 27.
- Misapplying the limit (1 + x/n)^{kn} without factoring out n can cause errors; proper factoring yields the cube of the basic limit.

Conclusion: The limit of {a_n} is 27, which matches the provided answer.

Suppose that an= ( 1 + ln ⁢3 𝑛 ) 3 𝑛 . What is the limit of the sequence \{a_n\}_{n=1}^\infty ? Type your answer in the box. It should be a whole number.简答题

类似问题

Let \(f(x) = x^2 \sin(x)\) and let \(x_n=1/n\). What is the limit of the sequence \(\{f(x_n)\}\)?

If the positive-value sequences \(\{x_n\}, \{y_n\}\) have limits 5 and 3 respectively, what are the limits of the sequences \(\{x_n+y_{n+1}\}\), \(\{x_ny_{2n}\}\), \(\{2x_n\}\) and \(\{x_n^{y_n}\} \).Enter your answer in the form ,,,

Let [math: f(x)=x2sin⁡(x)]f(x) = x^2 \sin(x) and let [math: xn=1/n]x_n=1/n. What is the limit of the sequence [math: {f(xn)}]\{f(x_n)\}?

Let [math: f(x)=x2]f(x) = x^2 and [math: xn=2+1/n]x_n=2+1/n.What is the limit of the sequence [math: {f(xn)}]\{f(x_n)\}?

If the positive-value sequences [math: {xn},{yn}]\{x_n\}, \{y_n\} have limits 5 and 3 respectively, what are the limits of the sequences [math: {xn+yn+1}]\{x_n+y_{n+1}\}, [math: {xny2n}]\{x_ny_{2n}\}, [math: {2xn}]\{2x_n\} and [math: {xnyn}]\{x_n^{y_n}\} .Enter your answer in the form ,,,

Let { 𝑎 𝑛 } 𝑛 = 0 ∞ be a sequence. Which statements are equivalent to " { 𝑎 𝑛 } 𝑛 = 0 ∞ is divergent to ∞ " ? Select all the correct answers.

How did Egyptian culture and governance change during the Ptolemaic period compared to the pharaonic period?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单