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Sequences and Series

  • Understand the concept of sequences and their limits.
  • Learn the definition of infinite series and when they converge.
  • Apply common convergence tests for series.
  • Explore power series and Taylor series.

Definition of Sequences

A **sequence** is an ordered list of numbers generated by a rule. A sequence is usually written as:

\[ a_1, a_2, a_3, \dots, a_n, \dots \]

For example, the sequence \( a_n = \frac{1}{n} \) is:

\[ 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots \]

Limits of Sequences

The **limit** of a sequence \( a_n \) is defined as:

\[ \lim_{n \to \infty} a_n = L \]

If this limit exists, the sequence **converges** to \( L \); otherwise, it diverges.

Definition of Series

A **series** is the sum of terms in a sequence:

\[ S_n = a_1 + a_2 + a_3 + \dots + a_n \]

It converges if the sum approaches a finite limit.

Convergence Tests

Some common tests to check if a series converges:

  • Comparison Test: If \( 0 \leq a_n \leq b_n \) and \( \sum b_n \) converges, then \( \sum a_n \) converges.
  • Ratio Test: If \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = r \):
    • \( r < 1 \Rightarrow \) Series converges.
    • \( r > 1 \Rightarrow \) Series diverges.
    • \( r = 1 \Rightarrow \) Inconclusive.
  • Integral Test: If \( f(x) \) is a decreasing, positive function and \( \int_1^\infty f(x)dx \) converges, then \( \sum a_n \) converges.

Power Series and Taylor Series

A **power series** is a series of the form:

\[ \sum_{n=0}^{\infty} c_n (x-a)^n \]

The **Taylor Series** for a function \( f(x) \) at \( x = a \) is:

\[ \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n \]

For example, the Taylor Series for \( e^x \) at \( x = 0 \) is:

\[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \]

Examples

Example 1: Determine if the series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) converges.

Using the **p-series test**, since \( p = 2 > 1 \), the series converges.

Exercises

  • Question 1: Find the limit of the sequence \( a_n = \frac{2n+3}{5n+4} \).
  • Question 2: Does the series \( \sum_{n=1}^{\infty} \frac{1}{n} \) converge?
  • Question 3: Use the ratio test to determine if \( \sum \frac{3^n}{n!} \) converges.

  • Answer 1: \( \lim_{n \to \infty} \frac{2n+3}{5n+4} = \frac{2}{5} \).
  • Answer 2: The harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) diverges.
  • Answer 3: Ratio test gives \( \lim_{n \to \infty} \frac{3^{n+1}/(n+1)!}{3^n/n!} = \frac{3}{n+1} \to 0 \), so the series converges.

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