Question

state whether the given series converges and explain why. (hint: rewrite using a change of index.)

\\\sum_{n = 1}^{\infty} \frac{1}{n + 1,100}\\

• the series converges because it is a geometric series where \\(-1 < r < 1\\).
• not enough information is given to determine whether the series converges or diverges.
• the series diverges because it is the harmonic series.
• the series converges because its the terms are decreasing.
• the series diverges because it is a geometric series where \\(r < -1\\).

Explanation:

Shift the index of summation

$$
LATEXBLOCK0
$$

Analyze the shifted series

$$
LATEXBLOCK1
$$

Answer:

• The series converges because it is a geometric series where \(-1 < r < 1\).
• Not enough information is given to determine whether the series converges or diverges.
• The series diverges because it is the harmonic series. (Correct answer)
• The series converges because its the terms are decreasing.
• The series diverges because it is a geometric series where \(r < -1\).