Kuzee
Sequences

Convergence Criteria

Determining whether a sequence converges and finding its limiting value are the core problems of limit theory. This section introduces several important criteria and commonly used limits.

Commonly Used Limits

Mastering these basic limits is very important for computing complex limits:

Basic limits
1.lim⁡n→∞1nk=0(k>0)2.lim⁡n→∞qn={0,∣q∣<11,q=1∞,q>1does not exist,q≤−13.lim⁡n→∞an=1(a>0)4.lim⁡n→∞nn=1\begin{aligned} &1. \quad \lim_{n \to \infty} \frac{1}{n^k} = 0 \quad (k > 0) \\ &2. \quad \lim_{n \to \infty} q^n = \begin{cases} 0, & |q| < 1 \\ 1, & q = 1 \\ \infty, & q > 1 \\ \text{does not exist}, & q \leq -1 \end{cases} \\ &3. \quad \lim_{n \to \infty} \sqrt[n]{a} = 1 \quad (a > 0) \\ &4. \quad \lim_{n \to \infty} \sqrt[n]{n} = 1 \end{aligned}

An Important Limit: The Natural Constant ee

The definition of e
lim⁡n→∞(1+1n)n=e≈2.71828⋯\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e \approx 2.71828\cdots

This is one of the most important constants in mathematics, with important applications in calculus, probability theory, compound interest calculations, and many other fields.

Generalized form:

lim⁡n→∞(1+kn)n=ek\lim_{n \to \infty} \left(1 + \frac{k}{n}\right)^n = e^k

The Ratio Test

For a positive-term sequence {an}\{a_n\}, if lim⁡n→∞an+1an=q\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = q, then:

  • If q<1q < 1, then lim⁡n→∞an=0\lim_{n \to \infty} a_n = 0
  • If q>1q > 1, then lim⁡n→∞an=∞\lim_{n \to \infty} a_n = \infty
  • If q=1q = 1, the test is inconclusive

Example: determine the limit of an=2nn!a_n = \frac{2^n}{n!}

an+1an=2n+1(n+1)!⋅n!2n=2n+1→0<1\frac{a_{n+1}}{a_n} = \frac{2^{n+1}}{(n+1)!} \cdot \frac{n!}{2^n} = \frac{2}{n+1} \to 0 < 1

Therefore lim⁡n→∞2nn!=0\lim_{n \to \infty} \frac{2^n}{n!} = 0

The Root Test

For a positive-term sequence {an}\{a_n\}, if lim⁡n→∞ann=q\lim_{n \to \infty} \sqrt[n]{a_n} = q, then:

  • If q<1q < 1, then lim⁡n→∞an=0\lim_{n \to \infty} a_n = 0
  • If q>1q > 1, then lim⁡n→∞an=∞\lim_{n \to \infty} a_n = \infty
  • If q=1q = 1, the test is inconclusive

Worked Examples

Example 1: Using the Important Limit

Find lim⁡n→∞(1+2n)n\lim_{n \to \infty} \left(1 + \frac{2}{n}\right)^n

Solution:

lim⁡n→∞(1+2n)n=lim⁡n→∞[(1+2n)n2]2=e2\lim_{n \to \infty} \left(1 + \frac{2}{n}\right)^n = \lim_{n \to \infty} \left[\left(1 + \frac{2}{n}\right)^{\frac{n}{2}}\right]^2 = e^2

Example 2: The Ratio Test

Determine lim⁡n→∞n102n\lim_{n \to \infty} \frac{n^{10}}{2^n}

Solution:

an+1an=(n+1)102n+1⋅2nn10=12⋅(n+1n)10=12⋅(1+1n)10→12<1\frac{a_{n+1}}{a_n} = \frac{(n+1)^{10}}{2^{n+1}} \cdot \frac{2^n}{n^{10}} = \frac{1}{2} \cdot \left(\frac{n+1}{n}\right)^{10} = \frac{1}{2} \cdot \left(1 + \frac{1}{n}\right)^{10} \to \frac{1}{2} < 1

Therefore lim⁡n→∞n102n=0\lim_{n \to \infty} \frac{n^{10}}{2^n} = 0

(Exponential growth is faster than polynomial growth)

Practice Problems

Exercise 1

Find lim⁡n→∞(1−1n)n\lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^n

Reference Answer(1 个标签)
convergence criteria of sequences

Solution:

lim⁡n→∞(1−1n)n=lim⁡n→∞(1+−1n)n=e−1=1e\lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^n = \lim_{n \to \infty} \left(1 + \frac{-1}{n}\right)^n = e^{-1} = \frac{1}{e}

Answer: 1e\frac{1}{e}

Exercise 2

Find lim⁡n→∞3nn!\lim_{n \to \infty} \frac{3^n}{n!}

Reference Answer(1 个标签)
convergence criteria of sequences

Solution:

Use the ratio test:

an+1an=3n+1(n+1)!⋅n!3n=3n+1→0<1\frac{a_{n+1}}{a_n} = \frac{3^{n+1}}{(n+1)!} \cdot \frac{n!}{3^n} = \frac{3}{n+1} \to 0 < 1

Therefore lim⁡n→∞3nn!=0\lim_{n \to \infty} \frac{3^n}{n!} = 0

Answer: 00

Exercise 3

Find lim⁡n→∞(n+1n)2n\lim_{n \to \infty} \left(\frac{n+1}{n}\right)^{2n}

Reference Answer(1 个标签)
convergence criteria of sequences

Solution:

lim⁡n→∞(n+1n)2n=lim⁡n→∞(1+1n)2n=lim⁡n→∞[(1+1n)n]2=e2\lim_{n \to \infty} \left(\frac{n+1}{n}\right)^{2n} = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{2n} = \lim_{n \to \infty} \left[\left(1 + \frac{1}{n}\right)^n\right]^2 = e^2

Answer: e2e^2

Summary

Symbols Used in This Article

符号类型读音/说明在本文中的含义
ee常数Euler’s numberThe natural constant, about 2.71828
n!n!阶乘n factorial1×2×⋯×n1 \times 2 \times \cdots \times n

中英对照

中文术语英文术语音标说明
自然常数natural constant/ˈnætʃrəl ˈkɒnstənt/The constant ee
比值判别法ratio test/ˈreɪʃiəʊ test/Determining convergence through the ratio of adjacent terms
根值判别法root test/ruːt test/Determining convergence through the nnth root
连续复利continuous compounding/kənˈtɪnjuəs kəmˈpaʊndɪŋ/The number of compounding periods tends to infinity
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