IB Mathematics AA SL The sum of an infinite geometric sequence Study Notes

Sum of an Infinite Geometric Sequence

An infinite geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a constant number called the common ratio. The sum of an infinite geometric sequence is the sum of all the terms in the sequence.

Formula for the Sum of an Infinite Geometric Sequence:

\( S_{\infty} = \frac{a}{1 – r} \)

where \( S_{\infty} \) is the sum of the infinite geometric sequence, \( a \) is the first term, and \( r \) is the common ratio.

Conditions for the Sum to Exist:

The sum of an infinite geometric sequence exists only if the absolute value of the common ratio is less than 1. In other words, \( |r| < 1 \).

Examples:

• The sum of the infinite geometric sequence \( 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots \) is 2.
• The sum of the infinite geometric sequence \( 3, -6, 12, -24, \ldots \) does not exist because the absolute value of the common ratio is greater than 1.

Proof of the Formula:

To prove the formula for the sum of an infinite geometric sequence, we can use the following steps:

Let \( S_n \) be the sum of the first \( n \) terms of the sequence.

Then, we have \( S_n = a + ar + ar^2 + \ldots + ar^{n-1} \).

Multiplying both sides of the equation by \( r \), we get \( rS_n = ar + ar^2 + ar^3 + \ldots + ar^n \).

Subtracting the second equation from the first equation, we get \( S_n – rS_n = a – ar^n \).

Factoring out \( S_n \) from the left-hand side, we get \( S_n(1 – r) = a – ar^n \).

Dividing both sides by \( (1 – r) \), we get \( S_n = \frac{a – ar^n}{1 – r} \).

Taking the limit as \( n \to \infty \), we get \( S_{\infty} = \lim_{n \to \infty} \frac{a – ar^n}{1 – r} = \frac{a}{1 – r} \).

Additional Notes:

• The sum of an infinite geometric sequence can also be expressed as an infinite series.
• The sum of an infinite geometric sequence is sometimes called the limit of the sequence.
• This sum can be used to solve various problems, such as finding the area of certain shapes or the total distance traveled by an object.