Given the series $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + ...$$ , what is the common ratio?

The common ratio r is $$\frac{1}{2}$$ .

Calculate the sum of the infinite geometric series $$\frac{3}{2} + \frac{3}{4} + \frac{3}{8} + ...$$ .

The sum S = a / (1 - r) = $$\frac{3/2}{1 - 1/2} = 3$$ .

Find the sum of the infinite geometric series $$\frac{5}{3} + \frac{5}{9} + \frac{5}{27} + ...$$ .

The sum S = $$\frac{5/3}{1 - 1/3} = \frac{5/3}{2/3} = \frac{5}{2} = 2.5$$ .

What is the first term of the series $$\frac{7}{2} + \frac{7}{4} + \frac{7}{8} + ...$$ ?

The first term a is $$\frac{7}{2}$$ .

If the first term of an infinite geometric series is 10 and the common ratio is $$\frac{1}{2}$$ , what is the sum?

The sum S = $$\frac{10}{1 - \frac{1}{2}} = 20$$ .

What is the significance of the first term in an infinite geometric series?

The first term 'a' is crucial as it determines the starting point of the series and influences the total sum.

Explain why the sum of the series $$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...$$ is equal to 2.

The series converges to 2 because S = a / (1 - r) = $$\frac{1}{1 - \frac{1}{2}} = 2$$ .

What is the sum of the infinite geometric series $$\frac{4}{5} + \frac{4}{25} + \frac{4}{125} + ...$$ ?

The sum S = $$\frac{4/5}{1 - \frac{1}{5}} = \frac{4/5}{4/5} = 1$$ .

How do you find the sum of an infinite geometric series with a negative common ratio?

The same formula applies: S = a / (1 - r), as long as |r|

Infinite Geometric Series Flashcards

Student preview

14 questions

Show all answers

1.

FLASHCARD QUESTION

Front

What is an infinite geometric series?

Back

An infinite geometric series is the sum of the terms of a geometric sequence that continues indefinitely. It converges if the absolute value of the common ratio is less than 1.

2.

FLASHCARD QUESTION

Front

What is the formula for the sum of an infinite geometric series?

Back

The sum S of an infinite geometric series can be calculated using the formula: S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio.

3.

FLASHCARD QUESTION

Front

What is the common ratio in a geometric series?

Back

The common ratio 'r' in a geometric series is the factor by which each term is multiplied to get the next term.

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