Ask a New Question
Menu

Which of the following sequences are convergent? Select all that apply.

(1 point)
Responses

a geometric sequence with r equals Start Fraction 1 over 5 End Fraction
a geometric sequence with Image with alt text: r equals Start Fraction 1 over 5 End Fraction

an arithmetic sequence with d equals negative 4
an arithmetic sequence with Image with alt text: d equals negative 4

a geometric sequence with r equals negative 2
a geometric sequence with Image with alt text: r equals negative 2

an arithmetic sequence with d equals Start Fraction 1 over 5 End Fraction
an arithmetic sequence with Image with alt text: d equals Start Fraction 1 over 5 End Fraction

a geometric sequence with r equals Start Fraction 2 over 3 End Fraction
a geometric sequence with Image with alt text: r equals Start Fraction 2 over 3 End Fraction

1 answer

To determine the convergence of the sequences listed, we need to analyze both geometric and arithmetic sequences.

  1. Geometric Sequence:

    • A geometric sequence converges if the absolute value of the common ratio \( r \) is less than 1 (\(|r| < 1\)).
    • If \( |r| \geq 1 \), the geometric sequence diverges.

  2. Arithmetic Sequence:

    • An arithmetic sequence diverges because it continues to increase or decrease without bound (the terms do not approach a fixed value).

Analysis of Given Sequences:

  1. a geometric sequence with \( r = \frac{1}{5} \):

    • \( |r| = \frac{1}{5} < 1 \)
    • Convergent

  2. an arithmetic sequence with \( d = -4 \):

    • This is an arithmetic sequence and it diverges.
    • Divergent

  3. a geometric sequence with \( r = -2 \):

    • \( |r| = 2 \geq 1 \)
    • Divergent

  4. an arithmetic sequence with \( d = \frac{1}{5} \):

    • This is an arithmetic sequence and it diverges.
    • Divergent

  5. a geometric sequence with \( r = \frac{2}{3} \):

    • \( |r| = \frac{2}{3} < 1 \)
    • Convergent

Summary of Convergence:

  • Convergent:

    • a geometric sequence with \( r = \frac{1}{5} \)
    • a geometric sequence with \( r = \frac{2}{3} \)

  • Divergent:

    • an arithmetic sequence with \( d = -4 \)
    • an arithmetic sequence with \( d = \frac{1}{5} \)
    • a geometric sequence with \( r = -2 \)

Thus, the sequences that are convergent are:

  • a geometric sequence with \( r = \frac{1}{5} \)
  • a geometric sequence with \( r = \frac{2}{3} \)