Question

which geometric series converges?

\\(\frac{1}{81} + \frac{1}{27} + \frac{1}{9} + \frac{1}{3} + \dots\\)

\\(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots\\)

\\(\sum_{n=1}^{\infty} 7(-4)^{n-1}\\)

\\(\sum_{n=1}^{\infty} \frac{1}{3}(2)^{n-1}\\)

Explanation:

State the convergence condition

Using the Geometric Series Convergence knowledge point
An infinite geometric series converges if and only if the absolute value of its common ratio \(r\) satisfies:

$$ |r| < 1 $$

Analyze the first option

Using the Common Ratio knowledge point

$$
LATEXBLOCK0
$$

Analyze the second option

Using the Common Ratio knowledge point

$$
LATEXBLOCK1
$$

Analyze the third option

Using the Common Ratio knowledge point

$$
LATEXBLOCK2
$$

Analyze the fourth option

Using the Common Ratio knowledge point

$$
LATEXBLOCK3
$$

Answer:

• (A) \(\frac{1}{81} + \frac{1}{27} + \frac{1}{9} + \frac{1}{3} + \dots\)
• (B) \(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots\) (Correct answer)
• (C) \(\sum_{n=1}^{\infty} 7(-4)^{n-1}\)
• (D) \(\sum_{n=1}^{\infty} \frac{1}{3}(2)^{n-1}\)