2.4 Limit Theorems for Sequences

Limit theorems for sequences are essential tools in mathematical analysis. They help us understand how sequences behave as they approach infinity, allowing us to manipulate and evaluate complex limits with ease.

These theorems, including the sum, product, and quotient rules, form the foundation for more advanced concepts in calculus. By mastering these rules, we can tackle intricate problems involving sequences and series, paving the way for deeper mathematical understanding.

Algebraic Limit Theorems for Sequences

Fundamental Algebraic Limit Theorems

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• The Sum Rule states that if $\lim(a_n) = L$ and $\lim(b_n) = M$, then $\lim(a_n + b_n) = L + M$
  • Example: If $\lim(a_n) = 3$ and $\lim(b_n) = 5$, then $\lim(a_n + b_n) = 3 + 5 = 8$
• The Difference Rule states that if $\lim(a_n) = L$ and $\lim(b_n) = M$, then $\lim(a_n - b_n) = L - M$
  • Example: If $\lim(a_n) = 7$ and $\lim(b_n) = 2$, then $\lim(a_n - b_n) = 7 - 2 = 5$
• The Constant Multiple Rule states that if $\lim(a_n) = L$ and $c$ is a constant, then $\lim(c * a_n) = c * L$
  • Example: If $\lim(a_n) = 4$ and $c = 3$, then $\lim(3 * a_n) = 3 * 4 = 12$
• The Product Rule states that if $\lim(a_n) = L$ and $\lim(b_n) = M$, then $\lim(a_n * b_n) = L * M$
  • Example: If $\lim(a_n) = 2$ and $\lim(b_n) = 6$, then $\lim(a_n * b_n) = 2 * 6 = 12$

Advanced Algebraic Limit Theorems

• The Quotient Rule states that if $\lim(a_n) = L$ and $\lim(b_n) = M$, where $M \neq 0$, then $\lim(a_n / b_n) = L / M$
  • Example: If $\lim(a_n) = 10$ and $\lim(b_n) = 2$, then $\lim(a_n / b_n) = 10 / 2 = 5$
• The Quotient Rule requires that the limit of the denominator sequence is non-zero to avoid division by zero
  • If $\lim(b_n) = 0$, the limit of the quotient sequence may not exist or may require further investigation using other techniques (L'Hôpital's Rule)
• The Power Rule states that if $\lim(a_n) = L$ and $p$ is a real number, then $\lim(a_n^p) = L^p$, provided that $L > 0$ if $p$ is not a rational number with an odd denominator
  • Example: If $\lim(a_n) = 4$ and $p = 1/2$, then $\lim(a_n^{1/2}) = 4^{1/2} = 2$

Applying Limit Theorems to Sequences

Evaluating Limits Using Algebraic Limit Theorems

• To evaluate the limit of a sequence using the Algebraic Limit Theorems, first identify the individual limits of the component sequences
  • Example: To find $\lim(3a_n - 2b_n)$, identify $\lim(a_n)$ and $\lim(b_n)$
• Apply the appropriate Algebraic Limit Theorem based on the operations involved in the sequence (addition, subtraction, multiplication, division, or constant multiplication)
  • Example: If $\lim(a_n) = 5$ and $\lim(b_n) = 3$, then $\lim(3a_n - 2b_n) = 3 \cdot \lim(a_n) - 2 \cdot \lim(b_n) = 3 \cdot 5 - 2 \cdot 3 = 9$
• Substitute the individual limits into the theorem to calculate the overall limit of the sequence

Limitations and Special Cases

• If the limit of a component sequence does not exist or violates the conditions of the theorem (division by zero), the limit of the entire sequence may not exist or may require further investigation
  • Example: If $\lim(a_n) = 4$ and $\lim(b_n) = 0$, then $\lim(a_n / b_n)$ is undefined due to division by zero
• In some cases, the limit of a sequence may exist even if the limits of its component sequences do not exist individually
  • Example: If $a_n = (-1)^n$ and $b_n = (-1)^{n+1}$, then $\lim(a_n)$ and $\lim(b_n)$ do not exist, but $\lim(a_n + b_n) = 0$
• Algebraic Limit Theorems can be combined to evaluate the limits of more complex sequences involving multiple operations
  • Example: To find $\lim(\frac{2a_n + 3b_n}{c_n - d_n})$, apply the Sum, Constant Multiple, and Quotient Rules

Monotone Convergence Theorem

Monotone Sequences and Boundedness

• The Monotone Convergence Theorem states that if a sequence is monotone (either non-decreasing or non-increasing) and bounded, then the sequence converges
• A sequence $(a_n)$ is non-decreasing if $a_n \leq a_{n+1}$ for all $n \in \mathbb{N}$, and non-increasing if $a_n \geq a_{n+1}$ for all $n \in \mathbb{N}$
  • Example: The sequence $(1 - \frac{1}{n})$ is non-decreasing because $1 - \frac{1}{n} \leq 1 - \frac{1}{n+1}$ for all $n \in \mathbb{N}$
• A sequence is bounded if there exist real numbers $m$ and $M$ such that $m \leq a_n \leq M$ for all $n \in \mathbb{N}$
  • Example: The sequence $(\frac{1}{n})$ is bounded because $0 \leq \frac{1}{n} \leq 1$ for all $n \in \mathbb{N}$

Applying the Monotone Convergence Theorem

• To apply the Monotone Convergence Theorem, first determine if the sequence is monotone by comparing consecutive terms
  • Example: To show that $(\frac{n}{n+1})$ converges, note that $\frac{n}{n+1} \leq \frac{n+1}{n+2}$ for all $n \in \mathbb{N}$, so the sequence is non-decreasing
• If the sequence is monotone, find the lower and upper bounds of the sequence
  • Example: For $(\frac{n}{n+1})$, we have $0 \leq \frac{n}{n+1} \leq 1$ for all $n \in \mathbb{N}$
• If both conditions are satisfied, the sequence converges. The limit of the sequence is equal to the supremum (for non-decreasing sequences) or the infimum (for non-increasing sequences) of the set of terms
  • Example: Since $(\frac{n}{n+1})$ is non-decreasing and bounded, it converges to its supremum, which is 1

Sequence Convergence Tests

Comparison Test

• The Comparison Test is used to determine the convergence or divergence of a sequence by comparing it to another sequence with known convergence properties
• If $0 \leq a_n \leq b_n$ for all $n \geq N$ (some $N \in \mathbb{N}$) and $\lim(b_n) = 0$, then $\lim(a_n) = 0$ (Squeeze Theorem)
  • Example: To show that $\lim(\frac{\sin n}{n}) = 0$, note that $0 \leq |\frac{\sin n}{n}| \leq \frac{1}{n}$ for all $n \geq 1$ and $\lim(\frac{1}{n}) = 0$
• If $a_n \leq b_n$ for all $n \geq N$ and $\sum b_n$ converges, then $\sum a_n$ converges
  • Example: To show that $\sum \frac{1}{n^2}$ converges, compare it to $\sum \frac{1}{n(n-1)}$, which converges by the p-series test
• If $a_n \geq b_n \geq 0$ for all $n \geq N$ and $\sum b_n$ diverges, then $\sum a_n$ diverges
  • Example: To show that $\sum \frac{1}{\sqrt{n}}$ diverges, compare it to $\sum \frac{1}{n}$, which diverges by the p-series test

Ratio Test

• The Ratio Test is used to determine the convergence or divergence of a series $\sum a_n$ by examining the limit of the ratio of consecutive terms, $\lim(|\frac{a_{n+1}}{a_n}|)$
  • If $\lim(|\frac{a_{n+1}}{a_n}|) < 1$, the series converges absolutely
    • Example: For $\sum \frac{2^n}{n!}$, $\lim(|\frac{2^{n+1}/(n+1)!}{2^n/n!}|) = \lim(\frac{2}{n+1}) = 0 < 1$, so the series converges absolutely
  • If $\lim(|\frac{a_{n+1}}{a_n}|) > 1$, the series diverges
    • Example: For $\sum n^2$, $\lim(|\frac{(n+1)^2}{n^2}|) = \lim(\frac{n^2+2n+1}{n^2}) = 1 > 1$, so the series diverges
  • If $\lim(|\frac{a_{n+1}}{a_n}|) = 1$, the test is inconclusive, and other tests should be used to determine convergence or divergence
    • Example: For $\sum \frac{1}{n}$, $\lim(|\frac{1/(n+1)}{1/n}|) = \lim(\frac{n}{n+1}) = 1$, so the Ratio Test is inconclusive (the series diverges by the p-series test)