Chapter 8

Explain why the sequence of partial sums for an alternating series is not an increasing sequence.

Explain how the Ratio Test works.

What is the defining characteristic of a geometric series? Give an example.

Give an example of a nonincreasing sequence with a limit.

Define sequence and give an example.

Describe how to apply the Alternating Series Test.

Is it true that if the terms of a series of positive terms decrease to zero, then the series converges? Explain using an example.

Explain how the Root Test works.

What is the difference between a geometric sum and a geometric series?

Give an example of a nondecreasing sequence without a limit.

Suppose the sequence \(\left\\{a_{n}\right\\}\) is defined by the explicit formula \(a_{n}=1 / n,\) for \(n=1,2,3, \ldots .\) Write out the first five terms of the sequence.

Why does the value of a converging alternating series with terms that are non increasing in magnitude lie between any two consecutive terms of its sequence of partial sums?

Can the Integral Test be used to determine whether a series diverges?

Explain how the limit Comparison Test works.

What is meant by the ratio of a geometric series?

Give an example of a bounded sequence that has a limit.

Suppose the sequence \(\left\\{a_{n}\right\\}\) is defined by the recurrence relation \(a_{n+1}=n a_{n},\) for \(n=1,2,3, \ldots,\) where \(a_{1}=1 .\) Write out the first five terms of the sequence.

Suppose an alternating series with terms that are non increasing in magnitude converges to a value \(L\). Explain how to estimate the remainder that occurs when the series is terminated after \(n\) terms.

For what values of \(p\) does the series \(\sum_{k=1}^{\infty} \frac{1}{k^{p}}\) converge? For what values of \(p\) does it diverge?

What is the first test you should use in analyzing the convergence of a series?

Does a geometric sum always have a finite value?

Give an example of a bounded sequence without a limit.

Define finite sum and give an example.

Explain why the magnitude of the remainder in an alternating series (with terms that are nonincreasing in magnitude) is less than or equal to the magnitude of the first neglected term.

For what values of \(p\) does the series \(\sum_{k=10}^{\infty} \frac{1}{k^{p}}\) converge (initial index is 10 )? For what values of \(p\) does it diverge?

What test is advisable if a series of positive terms involves a factorial term?

For what values of \(r\) does the sequence \(\left\\{r^{n}\right\\}\) converge? Diverge?

Define infinite series and give an example.

Explain why the sequence of partial sums for a series with positive terms is an increasing sequence.

What is the condition for convergence of the geometric series \(\sum_{k=0}^{\infty} a r^{k} ?\)

Explain how the methods used to find the limit of a function as \(x \rightarrow \infty\) are used to find the limit of a sequence.

Given the series \(\sum_{k=1}^{\infty} k\), evaluate the first four terms of its sequence of partial sums \(S_{n}=\sum_{k=1}^{n} k\)

Is it possible for a series of positive terms to converge conditionally? Explain.

Explain why, with a series of positive terms, the sequence of partial sums is an increasing sequence.

Evaluate each geometric sum. $$\sum_{k=0}^{8} 3^{k}$$

Compare the growth rates of \(\left\\{n^{100}\right\\}\) and \(\left\\{e^{n / 100}\right\\}\) as \(n \rightarrow \infty\).

The terms of a sequence of partial sums are defined by \(S_{n}=\sum_{k=1}^{n} k^{2}\) for \(n=1,2,3, \ldots .\) Evaluate the first four terms of the sequence.

Why does absolute convergence imply convergence?

If a series of positive terms converges, does it follow that the remainder \(R_{n}\) must decrease to zero as \(n \rightarrow \infty\) ? Explain.

Evaluate each geometric sum. $$\sum_{k=0}^{10}\left(\frac{1}{4}\right)^{k}$$

Explain how two sequences that differ only in their first ten terms can have the same limit.

Consider the infinite series \(\sum_{k=1}^{\infty} \frac{1}{k} .\) Evaluate the first four terms of the sequence of partial sums.

Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. $$\sum_{k=0}^{\infty} \frac{k}{2 k+1}$$

Evaluate each geometric sum. $$\sum_{k=0}^{20}\left(\frac{2}{5}\right)^{2 k}$$

Find the limit of the following sequences or determine that the limit does not exist. $$\left\\{\frac{n^{3}}{n^{4}+1}\right\\}$$

Write the first four terms of the sequence \(\left\\{a_{n}\right\\}_{n=1}^{\infty}\) $$a_{n}=1 / 10^{n}$$

Give an example of a series that converges conditionally but not absolutely.

Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. $$\sum_{k=1}^{\infty} \frac{k}{k^{2}+1}$$

Evaluate each geometric sum. $$\sum_{k=4}^{12} 2^{k}$$

Use the Ratio Test to determine whether the following series converge. $$\sum_{k=1}^{\infty} \frac{2^{k}}{k !}$$