Calculus Sequence and Series Problems

Practice Problems

  1. Finding an Explicit Formula for a Sequence

    Given the sequence 3, 7, 11, 15, 19, …, find an explicit formula to represent it.

    • Determine whether the sequence converges or diverges.

  2. Writing an Infinite Series in Sigma Notation

    Given the infinite series \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \frac{1}{243} + …, write it in sigma notation.

    • Determine whether the series converges or diverges.

  3. Using the Divergence Test

    Given the sequence an = \frac{n^2}{2n^2 + 1}, examine the infinite series \sum{n=1}^{\infty} an = \sum{n=1}^{\infty} \frac{n^2}{2n^2 + 1}.

    • Use the Divergence Test to determine if the series diverges.

Here's an explanation of the requested concepts:

  • Writing Explicit Formulas for Series: An explicit formula allows you to directly calculate any term of a sequence without needing to know the preceding terms. For example, in the sequence 3, 7, 11, 15, …, the explicit formula is a_n = 4n - 1.

  • Determining Convergence or Divergence of Explicit Formulas: A sequence converges if its terms approach a specific limit as n approaches infinity. Otherwise, it diverges. For example, the sequence with an = \frac{1}{n} converges to 0, while the sequence with an = n diverges.

  • Divergence Test: The Divergence Test states that if \lim{n \to \infty} an \neq 0, then the series \sum{n=1}^{\infty} an diverges. However, if \lim{n \to \infty} an = 0, the test is inconclusive.

  • Writing a Series Based on a Sequence: A series is the sum of the terms of a sequence. If you have a sequence a1, a2, a3, …, the corresponding series is a1 + a2 + a3 + …, which can be written in sigma notation as \sum{n=1}^{\infty} an.

  • Geometric Series: A geometric series is a series where each term is multiplied by a constant ratio r. It has the form a + ar + ar^2 + ar^3 + …, where a is the first term.

  • Writing an Infinite Sum of Geometric Series: An infinite geometric series can be written as \sum_{n=0}^{\infty} ar^n = a + ar + ar^2 + ar^3 + ….

  • Convergence or Divergence of a Geometric Series: A geometric series converges if the absolute value of the common ratio r is less than 1 (|r| < 1). Otherwise, it diverges.

  • Where a Geometric Series Converges To: If a geometric series converges (|r| < 1), it converges to the sum \frac{a}{1 - r}, where a is the first term.