Question

Choose your test Use the test of your choice to determine whether the following series converge. $$\sum_{k=1}^{\infty}(\sqrt[k]{k}-1)^{2 k}$$

Short answer

Explain. $$\sum_{k=1}^{\infty}(\sqrt[k]{k}-1)^{2 k}$$ The series converges according to the Root Test. The limit of the kth root of the series' absolute value is found to be less than 1, indicating absolute convergence.

Step-by-step solution

Apply the Root Test

The Root Test states that for a series \(\sum_{k=1}^{\infty}a_k\), if \(\limsup_{k\to\infty} \sqrt[k]{|a_k|}<1\), then the series converges absolutely. If \(\limsup_{k\to\infty} \sqrt[k]{|a_k|}>1\), then the series diverges. If the limit equals 1, the test is inconclusive. Our series is already in absolute form, so we can find the limit \(\lim_{k\to\infty} \sqrt[k]{(\sqrt[k]{k}-1)^{2 k}}\).

Simplify the Limit Expression

We will now simplify \(\sqrt[k]{(\sqrt[k]{k}-1)^{2 k}}\) before taking the limit. Using the exponent properties, we have \((a^{m})^{n} = a^{mn}\), so we can rewrite the expression as: $$\sqrt[k]{(\sqrt[k]{k}-1)^{2 k}} = ((\sqrt[k]{k}-1)^{2 k})^{\frac{1}{k}}$$ Now, applying the exponent properties, we can rewrite it as: $$((\sqrt[k]{k}-1)^{2 k})^{\frac{1}{k}} = (\sqrt[k]{k}-1)^{2 k\cdot\frac{1}{k}} = (\sqrt[k]{k}-1)^2$$ So the limit expression simplifies to \(\lim_{k\to\infty} (\sqrt[k]{k}-1)^2\).

Evaluate the Limit

Now that we have simplified the limit expression, evaluate it: $$\lim_{k\to\infty} (\sqrt[k]{k}-1)^2$$ As \(k→\infty\), we have \(\sqrt[k]{k}→1\), so the expression inside the limit becomes: $$\lim_{k\to\infty} (1-1)^2 = 0$$ From the result, we can infer that the original limit (after simplification) is less than 1.

Conclude Convergence

From the Root Test, since the limit of the \(k\)th root of the series' absolute value is less than 1, the series converges absolutely. Therefore, the given series: $$\sum_{k=1}^{\infty}(\sqrt[k]{k}-1)^{2 k}$$ converges according to the Root Test.

Key concepts

Root Test

The Root Test is a handy tool in the world of infinite series. It helps us decide if a series converges or diverges. To apply this test, we look at a series \(\sum_{k=1}^{\infty}a_k\). We calculate the limit \(\limsup_{k\to\infty} \sqrt[k]{|a_k|}\). Here's the trick with the Root Test:

• If the limit is less than 1, the series converges absolutely.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the test is inconclusive.

In our example, the series was \(\sum_{k=1}^{\infty}(\sqrt[k]{k}-1)^{2 k}\). We applied the Root Test by finding the limit of the expression \(\sqrt[k]{(\sqrt[k]{k}-1)^{2 k}}\). By simplifying, we found that this limit is indeed less than 1, confirming convergence.

Series Convergence

Series convergence is all about understanding if the sum of an infinite set of numbers results in a finite value. To determine whether a series converges, we deploy various convergence tests. In addition to the Root Test, other tests include:

• Ratio Test
• Integral Test
• Comparison Test
• Alternating Series Test

Each test has its unique conditions and applications. By choosing the right test, one can often simplify the problem.
Convergence can be absolute or conditional. A series converges absolutely if the series of absolute values converges. In our case, using the Root Test revealed that the series \(\sum_{k=1}^{\infty}(\sqrt[k]{k}-1)^{2 k}\) converges absolutely as the limit calculated was less than 1.

Limit Superior

The term "limit superior," often abbreviated as "lim sup," is a concept used to understand the behavior of a sequence at infinity.
When dealing with the Root Test, we often encounter \(\limsup_{k\to\infty} \sqrt[k]{|a_k|}\). This represents the greatest limit point of the subsequence of \(\sqrt[k]{|a_k|}\) values as \(k\) grows larger.Here’s why it’s useful:

• It helps measure the "ceiling" of a sequence.
• The "lim sup" captures the "essential growth rate" of a sequence.

In applying the Root Test, calculating \(\limsup_{k\to\infty} \sqrt[k]{|a_k|}\) tells us if our series is staying beneath the convergence threshold. For our series \((\sqrt[k]{k}-1)^{2 k}\), the value was less than 1, indicating convergence.