Problem 5

Question

Determine whether the series converges or diverges. It is possible to solve Problems 4 through 19 without the Limit Comparison, Ratio, and Root Tests. $\sum_{k=10}^{\infty} \frac{10}{k \sqrt{k}}$

Step-by-Step Solution

Verified

Answer

The series $\sum_{k=10}^{\infty} \frac{10}{k \sqrt{k}}$ converges.

1Step 1: Recognize the series

Recognize the series $\sum_{k=10}^{\infty} \frac{10}{k \sqrt{k}}$. This is a series of fractions where the numerator is the constant 10 and the denominator is the variable k under a square root.

2Step 2: Rewrite the series

Rewrite $\sum_{k=10}^{\infty} \frac{10}{k \sqrt{k}}$ as $\sum_{k=10}^{\infty} \frac{10}{k^{3/2}}$ to simplify.

3Step 3: Identify known series

Think of known series which this can be related to. A p-series with $p > 1$ is well known to be a convergent series. Here, we have $\sum_{k=10}^{\infty} \frac{10}{k^{3/2}}$ which is a p-series with $p = 3/2$, that is, $p > 1$.

4Step 4: Use the Comparison Test

By the Comparison Test (since for all $k$, $\frac{10}{k^{3/2}} \leq \frac{10}{k^{3/2}}$, and the series $\sum_{k=10}^{\infty} \frac{10}{k^{3/2}}$ (the series with greater terms) is convergent), the given series must also be convergent.

Key Concepts

p-seriesComparison Testconvergent series

p-series

A p-series is a type of mathematical series that takes the form $\sum_{k=1}^{\infty} \frac{1}{k^p}$, where $p$ is a positive constant. Understanding p-series helps determine whether certain kinds of series converge or diverge, which is crucial in calculus and mathematical analysis.
The behavior of a p-series heavily depends on the value of $p$:

If $p > 1$, the series is convergent. This means that as we sum the terms of the series from 1 to infinity, the total sum approaches a specific finite number.
If $p \leq 1$, the series is divergent, implying that adding the series terms endlessly will not settle towards a single finite value, instead increasing without bound.

A common example of a p-series is the harmonic series, $\sum_{k=1}^{\infty} \frac{1}{k}$, which is divergent since $p = 1$. In the given problem, we are dealing with $\sum_{k=10}^{\infty} \frac{10}{k^{3/2}}$, which can be seen as a p-series where $p = 3/2$. Since $3/2 > 1$, we can conclude that this particular series is convergent.

Comparison Test

The Comparison Test is a valuable tool in determining the convergence or divergence of series. This test compares a given series with a second series whose convergence behavior is already known.
To apply the Comparison Test effectively:

Identify a series that is close in form to the original series with known convergence behavior.
If the series you know is convergent and has terms greater than those of the given series, then the given series is also convergent.
Conversely, if the known series is divergent and has terms smaller than those of the given series, the given series is divergent as well.

In our exercise, we compared $\sum_{k=10}^{\infty} \frac{10}{k^{3/2}}$ to itself, since we had already judged it as a convergent p-series. According to the Comparison Test, because this series is convergent, any series with smaller terms must also be convergent. This means that the original series $\sum_{k=10}^{\infty} \frac{10}{k \sqrt{k}}$ converges.

convergent series

In mathematical terms, a convergent series is one where the sum of its terms approaches a specific finite value as more terms are added. This finite value is known as the 'limit' of the series. Convergence is a fundamental concept because it reveals whether continually adding terms will lead to a stable sum or not.
The basic properties of a convergent series include:

The series has a finite sum; it does not go to infinity.
The partial sums of the series stabilizes, meaning adding more terms changes the sum very little, consistently shrinking to zero.

Knowing that $\sum_{k=10}^{\infty} \frac{10}{k^{3/2}}$ is convergent implies that the addition of all terms from $k = 10$ to infinity converges to a certain limit.
When a series converges, it assures us of its mathematical completeness and stability, providing powerful insight into the behavior of functions or data that may be modeled or represented by such a series.

Problem 4

Problem 5

Other exercises in this chapter

Problem 4

Find the Taylor series for $f(x)$ centered at the indicated value of $b$. $$ f(x)=\sin x, \quad b=\pi $$

View solution

Problem 4

Do the following. (a) Compute the fourth degree Taylor polynomial for $f(x)$ at $x=0 .$ (b) On the same set of axes, graph \(f(x), P_{1}(x), P_{2}(x), P_{3}

View solution

Problem 5

For each series, determine whether the series converges absolutely, converges conditionally, or diverges. $$ \sum_{k=10}^{\infty} \frac{\cos (k n)}{10 k} $$

View solution

Problem 5

The second degree Taylor polynomial for $f(x)=(1+x)^{p}$ is $1+p x+\frac{p(p-1)}{2 !} x^{2} .$ If the second degree Taylor polynomial is used to approximate

View solution