Question

We know from Section 8.3 that the geometric series \(\sum a r^{k}(a \neq 0)\) converges if \(01 .\) Prove these facts using the Integral Test, the Ratio Test, and the Root Test. Now consider all values of \(r\) What can be determined about the geometric series using the Divergence Test?

Short answer

In conclusion: We analyzed a geometric series \(\sum a r^k\) using four tests: 1) The Integral Test: The series converges if \(01\). 2) The Ratio Test: The series converges if \(01\). 3) The Root Test: The series converges if \(01\). 4) The Divergence Test: We could not determine the convergence or divergence of the series for all values of \(r\). Overall, the Integral, Ratio, and Root Tests each concluded that the geometric series converges for \(01\). The Divergence Test could not provide a general conclusion for all values of \(r\).

Step-by-step solution

Apply the Integral Test

Firstly, let us analyze the given series using the Integral Test. The Integral Test states that if \(f(k) = a r^k\) is positive, continuous, and decreasing on an interval, then the series \(\sum a r^k\) converges if the integral \(\int^{\infty}_{1} f(k) dk\) converges and diverges if the integral diverges. Let's compute this integral: $$ \int^{\infty}_{1} a r^k dk = a \int^{\infty}_{1} r^k dk = a \left [ \frac{r^k}{\ln r} \right ]^{+\infty}_{1} $$ This integral converges if \(01\) confirming the geometrical series behavior is consistent with the exercise statement.

Apply the Ratio Test

Now, let's use the Ratio Test to analyze the geometric series. The Ratio Test states that if the limit of the ratios between consecutive terms, \(L = \lim_{k\to\infty} |\frac{a_{k+1}}{a_k}|\), exists, then the series converges if \(L < 1\) and diverges if \(L > 1\). Let's compute this limit: $$ L = \lim_{k\to\infty} \frac{a r^{k+1}}{a r^k} = \lim_{k\to\infty} r = r $$ This limit converges if \(01\), which confirms the geometrical series behavior asserted in the exercise statement.

Apply the Root Test

Next, let us analyze the given series using the Root Test. The Root Test states that if the limit of the nth root of the absolute value of the terms, \(L = \lim_{k\to\infty} |a_k|^{\frac{1}{k}}\), exists, then the series converges if \(L < 1\) and diverges if \(L > 1\). Let's compute this limit: $$ L = \lim_{k\to\infty} |a r^k|^{\frac{1}{k}} = \lim_{k\to\infty} |a|^{\frac{1}{k}} |r| = |r| $$ This limit converges if \(01\). Again, this confirms the geometrical series behavior mentioned in the exercise statement.

Apply the Divergence Test

Finally, let's apply the Divergence Test to the geometric series. The Divergence Test states that if the limit of the terms of a series, \(\lim_{k\to\infty} a_k\), is not equal to zero, then the series diverges. Let's compute this limit: $$ \lim_{k\to\infty} a r^k = a \lim_{k\to\infty} r^k $$ We cannot determine the convergence or divergence of the series using the Divergence Test for all values of \(r\) as the limit depends on the value of \(r\).

Key concepts

Integral Test

Understanding the Integral Test can be essential when studying series convergence. For a series \(\sum a_k\), where \(a_k = f(k)\) for a function \(f\) that is positive, continuous, and decreasing for \(k \geqslant n_0\) (some starting index), the Integral Test helps determine convergence. If the improper integral \(\int_{n_0}^{\infty} f(k)dk\) converges, so does the series, and if the integral diverges, so does the series.

In the case of geometric series, the function \(f(k) = ar^k\) fits the criteria for the Integral Test as long as \(0 1\), both the integral and the series diverge. Thus, we confirm the behavior of a geometric series through the Integral Test.

Ratio Test

The Ratio Test is another tool that offers a way to tackle series convergence, particularly helpful with series where the terms involve exponentials, factorials, or powers. By comparing the size of consecutive terms, the Ratio Test is defined as \(L = \lim_{k \to \infty} |\frac{a_{k+1}}{a_k}|\).

A geometric series \(\sum ar^k\) can be examined using the Ratio Test. When applying it to \(ar^k\), \(L\) becomes the constant ratio \(r\), and no limit calculation is necessary. If \(0 < r < 1\), \(L < 1\), which means the series converges. Conversely, if \(r > 1\), \(L > 1\), indicating that the series diverges. This test reaffirms the geometric series' properties and provides quick convergence or divergence evaluation.

Root Test

Like the Ratio Test, the Root Test is particularly valuable for series with terms involving roots or powers. It focuses on the nth root of the absolute value of each term, with the general form \(L = \lim_{k \to \infty} |a_k|^\frac{1}{k}\). The series converges if \(L < 1\) and diverges if \(L > 1\).

In analyzing a geometric series through the Root Test, we raise each term \(ar^k\) to the \(1/k\) power and take the limit as \(k\) approaches infinity. The result simplifies to \(r\) given that \(a\) is a non-zero constant. Since \(|a|^\frac{1}{k}\) approaches 1 as \(k\) increases, \(L = |r|\). This reaffirms the conditions for convergence (\(0 1\)) of geometric series.

Divergence Test

The Divergence Test is the simplest of the convergence tests, used as a preliminary tool. If the limit of the terms \(\lim_{k \to \infty} a_k\) is not zero, the series cannot converge. However, the inverse is not necessarily true; a limit of zero does not guarantee convergence.

When we apply this test to a geometric series, we examine the limit of \(ar^k\) as \(k\) approaches infinity. If \(r > 1\), the term grows without bound, and the series diverges. For \(0 < r < 1\), the term shrinks and the limit is zero, which is inconclusive as the test doesn't confirm convergence. Thus, while the Divergence Test can instantly reveal divergence for \(r > 1\), it cannot confirm convergence for a geometric series when \(0