Problem 37 - Algebra 2

Question

Evaluate each infinite series that has a sum. $$ \sum_{n=1}^{\infty} 7(2)^{n-1} $$

Step-by-Step Solution

Answer

The given series is divergent and so it doesn't have a sum.

Step 1: Identify the Common Ratio

In the given series, \( \sum_{n=1}^{\infty} 7(2)^{n-1} \), we need to identify the common ratio (r). The common ratio of a geometric series is found by dividing any term by the previous term. In this series, since the term is of the form 7*(2^{n-1}), the common ratio (r) will be 2.

Step 2: Test for Convergence/Divergence

Now, determine if the series converges or diverges. A geometric series converges if the absolute value of the common ratio |r| is less than 1, and diverges if |r| is greater than or equal to 1. In this case, the |r|=2, which is greater than 1, so this series diverges.

Step 3: Sum of the Series

Since we found that the series diverges, it does not have a sum. So there is no sum for the given series.

Key Concepts

Infinite Series

An infinite series is a sum of an endless sequence of numbers. It often starts from a certain number and extends to infinity. In mathematical terms, an infinite series takes the form

• \( a_1 + a_2 + a_3 + ext{...} \)

where the series keeps going forever.
Infinite series can be overwhelming, but they are essential in various mathematical applications.
Understanding when and how these series add up to a finite sum, or if they do at all, is the cornerstone of series analysis in calculus.
Not all infinite series have a sum. Only convergent series, where the terms approach zero, may form a complete sum. Though called infinite, these series can represent finite values under certain conditions.
In such cases, finding the sum involves using formulas specifically designed for these converging series.

Convergence

Convergence in the context of an infinite series signifies that as you add more and more terms, their sum approaches a specific finite value.

• A series is said to be convergent if the terms get smaller and smaller.

When you're dealing with geometric series, a special condition is applied to check for convergence.
The absolute value of the common ratio \(|r|\) must be less than 1.
This means that its terms keep reducing, minimizing the overall contribution to the sum as you move further along the series.
If \(|r|\) is equal to or greater than 1, the series diverges, meaning it will not settle on a single, finite sum.
Convergence is crucial because it tells us if finding the sum of an infinite series is possible at all.

Common Ratio

The common ratio is a fundamental component of a geometric series. It is found by dividing any term in the series by the preceding term.

• For instance, in the series \( 2, 4, 8, 16, \ldots \), the common ratio \( r \) would be \( \frac{4}{2} = 2 \).

It's constant throughout the series, determining the nature of its progression.
In our original exercise, where the series is \( \sum_{n=1}^{\infty} 7(2)^{n-1} \),
the common ratio is readily identified as 2.
Understanding the common ratio is essential because it helps in checking whether the series will converge or diverge.
For geometric series, this ratio directly influences the divergence or convergence, making it an important concept to grasp.