Evaluating Infinite Geometric Series

Front

What is a geometric series?

Back

A geometric series is the sum of the terms of a geometric sequence, where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

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How do you find the sum of the first n terms of a geometric series?

Back

The sum of the first n terms (S_n) of a geometric series can be calculated using the formula: S_n = a(1 - r^n) / (1 - r), where a is the first term and r is the common ratio.

Front

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

Back

The sum of an infinite geometric series exists if the absolute value of the common ratio r is less than 1. The formula is S = a / (1 - r), where a is the first term.

Front

What condition must be met for an infinite geometric series to converge?

Back

For an infinite geometric series to converge, the absolute value of the common ratio (|r|) must be less than 1.

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Back

Does Not Exist (the series diverges because the common ratio is greater than 1).

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Back

Front

What is the common ratio in a geometric series?

Back

The common ratio (r) is the factor by which each term is multiplied to get the next term in the series.

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How do you determine the common ratio of a geometric series?

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The common ratio can be determined by dividing any term by the previous term: r = a_n / a_(n-1).

Front

What is the first term of the series 2, 8, 32, 128,...?

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The first term (a) is 2.

Front

Find the sum of the first 6 terms of the geometric sequence 2, 8, 32, 128,...

Back

The sum is 2730.