Question

Determine whether the following statements are true and give an explanation or counterexample. a. \(\sum_{k=1}^{\infty}\left(\frac{\pi}{e}\right)^{-k}\) is a convergent geometric series. b. If \(a\) is a real number and \(\sum_{k=12}^{\infty} a^{k}\) converges, then \(\sum_{k=1}^{\infty} a^{k}\) converges. If the series \(\sum_{k=1}^{\infty} a^{k}\) converges and \(|a|<|b|,\) then the series \(\sum_{k=1}^{\infty} b^{k}\) converges. d. Viewed as a function of \(r,\) the series \(1+r^{2}+r^{3}+\cdots\) takes on all values in the interval \(\left(\frac{1}{2}, \infty\right)\) e. Viewed as a function of \(r,\) the series \(\sum_{k=1}^{\infty} r^{k}\) takes on all values in the interval \(\left(-\frac{1}{2}, \infty\right)\)

Short answer

Question: Determine which of the following statements are true. a) The series \(\sum_{k=1}^{\infty}\left(\frac{\pi}{e}\right)^{-k}\) converges. b) If the series \(\sum_{k=12}^{\infty} a^{k}\) converges, then the series \(\sum_{k=1}^{\infty} a^{k}\) converges. c) If the series \(\sum_{k=1}^{\infty} a^{k}\) converges, then the series \(\sum_{k=1}^{\infty} b^{k}\) converges for \(|a|<|b|\). d) The series \(1+r^{2}+r^{3}+\cdots\) takes on all values in the interval \(\left(\frac{1}{2}, \infty\right)\) as a function of \(r\). e) The series \(\sum_{k=1}^{\infty} r^{k}\) takes on all values in the interval \(\left(-\frac{1}{2}, \infty\right)\) as a function of \(r\).

Step-by-step solution

Statement a:

To determine whether the series \(\sum_{k=1}^{\infty}\left(\frac{\pi}{e}\right)^{-k}\) is convergent, we need to check if it is a geometric series and find the common ratio. A geometric series is a series of the form \(\sum_{k=1}^{\infty}ar^{k-1}\), where \(a\) is the first term and \(r\) is the common ratio between consecutive terms. In our case, \(\left(\frac{\pi}{e}\right)^{-k}\) is a geometric series with \(a = 1\) and \(r = \frac{e}{\pi}\). The geometric series converges if \(|-1<r<1|\), and as \(\frac{e}{\pi}<1\), the series converges. So statement a is true.

Statement b:

We are given that the series \(\sum_{k=12}^{\infty} a^{k}\) converges. In order to conclude that the series \(\sum_{k=1}^{\infty} a^{k}\) converges as well, we need to prove that the convergence doesn't depend on the starting point of the sum. If the series converges, |a| < 1, and so adding or removing a finite number of terms does not affect convergence. Since we are given that the series \(\sum_{k=12}^{\infty} a^{k}\) converges, we can conclude that the series \(\sum_{k=1}^{\infty} a^{k}\) converges as well. So statement b is true.

Statement c:

We know that the series \(\sum_{k=1}^{\infty} a^{k}\) converges and \(|a|<|b|\). To determine whether the series \(\sum_{k=1}^{\infty} b^{k}\) converges, we need to look at the convergence of a geometric series. We know that if the series converges, then \(|-1 |a|\). Given that the series \(\sum_{k=1}^{\infty} a^{k}\) converges, we can infer that \(|-1<a<1|\), but we cannot determine the convergence of \(\sum_{k=1}^{\infty} b^{k}\) under these conditions. For example, consider \(a = \frac{1}{2}\) and \(b = \frac{3}{2}\). The series \(\sum_{k=1}^{\infty} a^{k}\) converges, but \(\sum_{k=1}^{\infty} b^{k}\) does not. So statement c is false.

Statement d:

We want to determine whether the series \(1+r^{2}+r^{3}+\cdots\) takes on all values in the interval \(\left(\frac{1}{2}, \infty\right)\) as a function of \(r\). We can rewrite the given series as \(\sum_{k=0}^{\infty} r^{k+1}\). The given series converges if \(0<|r|<1\). Now, let \(S = 1+r^{2}+r^{3}+\cdots\). This is equivalent to \(S = \frac{r^2}{1-r}\). As \(r\) ranges from \(0\) to \(1\), the function \(S\) takes values from \(0\) to \(\infty\), not including the interval \(\left(\frac{1}{2}, \infty\right)\). Therefore, statement d is false.

Statement e:

We want to determine whether the series \(\sum_{k=1}^{\infty} r^{k}\) takes on all values in the interval \(\left(-\frac{1}{2}, \infty\right)\) as a function of \(r\). The given series converges if \(|-1<r<1|\). We can find a closed form for the given series: \(S = \frac{r}{1-r}\). As \(r\) ranges from \(-1\) to \(1\), the function \(S\) takes values from \(-\infty\) to \(\infty\), which includes the interval \(\left(-\frac{1}{2}, \infty\right)\). Therefore, statement e is true. In summary, statements a, b, and e are true, while statements c and d are false.

Key concepts

Geometric Series

Imagine a sequence of numbers where each term is multiplied by the same constant to get the next term. This is the essence of a geometric series. For example, if we start with the number 1 and multiply by 2 to get the next term, we get the sequence 1, 2, 4, 8, and so on. In mathematical notation, a geometric series is expressed as ewlineewlineewline \[\sum_{k=1}^\infty ar^{k-1}\] ewlineewline where \(a\) is the first term and \(r\) is the common ratio, the constant we multiply by to get the next term. The series adds up these terms from \(k=1\) to infinity. Geometric series can both converge or diverge, which depends on the value of the common ratio. If the absolute value of the common ratio is less than 1 (\(|r|<1\)), the series converges; if it is equal to or greater than 1, the series diverges.

Series Convergence

Determining whether a series converges is like asking if a long journey finally reaches a destination. For a geometric series, we have a simple rule to test for convergence: if the common ratio, \(r\), in absolute terms is less than 1, the series converges. This means the terms get smaller and smaller, eventually adding up to a finite number.

In contrast, when \(|r|\geq1\), the series diverges because the terms do not approach zero and their sum stretches to infinity. Convergence is significant because it allows us to use the series to calculate a precise sum, whereas a divergent series does not sum to a definitive value, continually growing without bound.

Sum of a Series

Understanding the sum of a series is key to grasping the notion of limits and infinite sequences. For a convergent geometric series, we can actually calculate the sum using a simple formula: \[S = \frac{a}{1-r}\] ewlineewline, where \(a\) is the first term and \(r\) is the common ratio. It's fascinating to think that adding up an infinite number of terms can result in a finite sum! The true power of this formula lies in its ability to provide a clear and precise value for what might seem to be an endless process.

Common Ratio

The common ratio, \(r\), is at the heart of the geometric series. It's the multiplier that connects one term to the next.

It's critical to note that the common ratio determines the behavior of the series: if the ratio is between -1 and 1 (not including -1 and 1), the series converges, and its terms shrink in size. If the ratio is exactly 1, the series remains constant and doesn't converge to a finite limit, and if the ratio is greater than 1 in absolute value, the series terms grow infinitely, indicating divergence. In short, the common ratio's value is indispensable in understanding and working with geometric series.