Question

If \(|r|<1,\) the sum of the geometric series \(\sum_{k=1}^{\infty} a r^{k-1}\) is _____ .

Short answer

The sum is \(\frac{a}{1 - r}\).

Step-by-step solution

Identify the geometric series

Recognize that the given series \(\sum_{k=1}^{\infty} a r^{k-1}\)is an infinite geometric series where each term is given by \(a r^{k-1}\).

Use the formula for the sum of an infinite geometric series

Recall that the sum of an infinite geometric series \(\sum_{k=0}^{\infty} ar^k\) is given by \(\frac{a}{1 - r}\) if \(|r| < 1\).

Adjust the series to the standard form

Notice that our series starts at \(k=1\) instead of \(k=0\). Rewrite the sum starting from \(k=0\) with the same general form: \(\sum_{k=1}^{\infty} a r^{k-1} = a r^{0} + a r^1 + a r^2 + \dots = a + ar + ar^2 + \dots\). This series is equivalent to the series \(ar^k\) starting from \(k=0\).

Apply the sum formula

Apply the formula for the sum of an infinite geometric series: \[\sum_{k=1}^{\infty} a r^{k-1} = a \sum_{k=0}^{\infty} r^k = \frac{a}{1 - r}.\]

Key concepts

Infinite Series

An infinite series is the sum of infinitely many terms. Think of it like a sequence of numbers that you keep adding on and on without end. The special thing about many infinite series, especially in mathematics, is that they can sum to a finite value, even though there are infinitely many terms. In the case of geometric series, an infinite series can converge to a number if the common ratio, denoted by \(r\), meets a specific condition.

Sum Formula

The sum formula is a mathematical expression that helps you find the total value of an infinite series. For the infinite geometric series, if the absolute value of the common ratio \(\left| r \right| < 1 \), then the series converges to a sum. This formula is:
\[\text{Sum} = \frac{a}{1 - r}\]
Here, 'a' is the first term of the series, and 'r' is the common ratio. This sum formula is powerful. It lets you compute the total sum of an infinite number of terms quickly and easily. Note that the condition \(\left| r \right| < 1 \) is essential for this formula to work.

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, in the sequence 2, 4, 8, 16, the common ratio is 2. If you keep multiplying by the common ratio, you get an infinite geometric sequence. We summarize it with this notation:
\(\text{Geometric Series} = \sum_{k=0}^{\infty} ar^k\)
This means you're adding up all the terms like this:

• First term: \(a\)
• Second term: \(ar\)
• Third term: \(ar^2\)
• ...and so on.

The common ratio \(r\) decides the growth of the series.

Convergence Criterion

The convergence criterion is what determines whether the infinite series sums to a finite number. For geometric series, this criterion is based on the common ratio \(r\). The series \(\sum_{k=0}^{\infty} ar^k\) converges only if the absolute value of the common ratio is less than 1, i.e., \(\left| r \right| < 1\).
Why is this important? Because if \(\left| r \right| \) is 1 or greater, the terms get too large, and the series does not settle down to a finite sum. When \(\left| r \right| < 1\), each successive term gets smaller and smaller, eventually becoming negligible. This makes the sum finite, allowing us to use the sum formula: \[\text{Sum} = \frac{a}{1 - r}\]