Problem 60 - A Graphical Approach to College Algebra

Question

Find each sum that converges. $$\sum_{k=1}^{\infty}(0.1)^{k}$$

Step-by-Step Solution

Answer

The sum is \(\frac{1}{9}\).

Step 1: Identify the Type of Series

The series is given as \(\sum_{k=1}^{\infty}(0.1)^{k}\). This is a geometric series where the first term \(a\) is 0.1 and the common ratio \(r\) is also 0.1.

Step 2: Check for Convergence

For a geometric series \(\sum_{k=1}^{\infty} ar^k\), it converges if and only if the absolute value of the common ratio \(|r| < 1\). Here, \(|0.1| < 1\), so the series converges.

Step 3: Calculate the Sum of the Series

The formula for the sum \(S\) of an infinite geometric series \(\sum_{k=1}^{\infty} ar^k\) is \(S = \frac{a}{1-r}\). Substitute \(a = 0.1\) and \(r = 0.1\) into the formula to find: \[S = \frac{0.1}{1 - 0.1} = \frac{0.1}{0.9} = \frac{1}{9}\].

Step 4: Final Result

Hence, the sum of the series \(\sum_{k=1}^{\infty}(0.1)^k\) is \(\frac{1}{9}\).

Key Concepts

Convergence

Convergence in the context of series, especially a geometric series, refers to the behavior of the series as the number of terms tends to infinity. When we deal with infinite series, it's crucial to determine whether they add up to a finite number—this is known as convergence. For a geometric series to converge, the absolute value of its common ratio \(|r|\) must be less than 1. This condition ensures that the added terms decrease in value, steering closer to zero, and thus the sum approaches a fixed finite limit.

• If \(|r| < 1\), each subsequent term becomes smaller, making the series converge.
• If \(|r| \geq 1\), the terms do not decrease consistently, leading to divergence.

In our exercise, the geometric series has a common ratio of 0.1, which satisfies the convergence criteria, as \(|0.1| < 1\). Hence, this series converges to a particular sum.

Infinite Series

An infinite series is essentially the sum of infinitely many terms. In mathematical terms, it's denoted with the summation symbol and limits extending infinitely. Understanding infinite series involves analyzing if adding an infinite number of terms leads to a meaningful result, such as a specific number.
In our example, \[\sum_{k=1}^{\infty}(0.1)^{k}\]represents a geometric series that includes an unbounded number of terms. Unlike finite series that stop at a certain number of terms, infinite series require the understanding of convergence to conclude about their sums. Infinite series can behave differently depending on the value of their terms:

• They converge to a number when the terms shrink towards zero.
• They diverge when the terms remain significant or grow larger.

For successful convergence analysis of an infinite series, identifying the series type and the common ratio in geometric series, like in this case, is crucial.

Sum of Series

The sum of an infinite series, when it converges, can often be found using specific formulas depending on the type of series. For geometric series, this calculation has a straightforward process. Once convergence is confirmed, we can use the formula:\[S = \frac{a}{1-r}\]where \(a\) is the first term, and \(r\) is the common ratio.

• It's imperative that \(|r|<1\) for the formula to apply as only under this condition the series converges.
• This formula simplifies the process by providing a direct means to find the sum without manual addition of terms.

Using this formula for our series, where the first term \(a\) is 0.1 and the common ratio \(r\) is 0.1, the sum can be calculated directly:\[S = \frac{0.1}{1 - 0.1} = \frac{0.1}{0.9} = \frac{1}{9}\]This result, \(\frac{1}{9}\), represents the sum of an infinite sequence of diminishing terms. It underscores how even an infinite series can converge to a specific value under the right conditions.