Determining Convergence or Divergence of a Series

Key Concepts

• Sequence vs. Series:

  • A sequence (a_n) is a list of numbers, whereas a series is the sum of a sequence up to a certain term.
  • Example: For (a_n = 2n), the sequence is (2, 4, 6, 8, \ldots).
  • Sums of sequences are denoted (S_n), where (S_1 = 2), (S_2 = 6), (S_3 = 12), etc.

• Infinite Series:

  • Sum (S_\infty) represents the sum of an infinite number of terms.
  • To determine if it converges or diverges, find (\lim_{n \to \infty} S_n).
  • If this limit exists and is finite, the series converges; otherwise, it diverges.

Determining Convergence/Divergence

• Convergent Series:

  • If (\lim_{n \to \infty} a_n = c) (a constant), the sequence converges.

• Divergent Series:

  • If the limit does not exist or goes to infinity, the sequence diverges.
  • If (\lim_{n \to \infty} S_n = \infty), the series diverges.

Example Analysis

• Example Series: (a_n = 2n)

  • Sequence: (a_1 = 2, a_2 = 4, a_3 = 6, \ldots)
  • Sum: As terms add up, results in infinity.
  • Therefore, the series diverges.

• General Equation for Partial Sums:

  • Arithmetic series formula: (S_n = \frac{n(a_1 + a_n)}{2}).
  • For (a_n = 2n), (S_n = n(n+1)).
  • (\lim_{n \to \infty} S_n = \infty) confirms divergence._

Divergence Test

• Definition: If (\lim_{n \to \infty} a_n \neq 0), the series diverges.

• Application Example:

  • For (a_n = 2n), (\lim_{n \to \infty} 2n \neq 0), thus diverges.

Additional Example

• Sequence: (a_n = \frac{5n+3}{7n-4})
  • Find (\lim_{n \to \infty} a_n = \frac{5}{7}), which is not 0.
  • Therefore, series diverges._

Conclusion

• Convergent Series:

  • Requires (\lim_{n \to \infty} a_n = 0).
  • Continual addition of values close to zero keeps sum finite.

• Divergent Series:

  • As (\lim_{n \to \infty} a_n \neq 0), sum trends to infinity.

Summary:

• Use limits to test convergence or divergence.
• Divergence test gives quick indication.
• Remember, a sequence can converge while its series may diverge.