Chapter 8, Problem 7RCC

a.

To determine

To find: if a series is convergent by the Integral Test, how do estimate its sum.

Answer to Problem 7RCC

  $s_n+{\displaystyle \underset{n+1}{\overset{\infty }{\int }}{a}_n}\le R_n\le {\displaystyle \underset{n}{\overset{\infty }{\int }}{a}_n}$ .

Explanation of Solution

Given information:

Calculation:

If the given series is ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ :

Then let $(a_n)=f(x),$ if the following integral converges, so does the series:

  ${\displaystyle \underset{1}{\overset{\infty }{\int }}f(x)dx}$ .

As for the partial sum, can find it by adding the first few terms, the number of how many term add.

Is determined from the Reminder Theorem:

  ${\displaystyle \underset{n+1}{\overset{\infty }{\int }}{a}_n}\le R_n\le {\displaystyle \underset{n}{\overset{\infty }{\int }}{a}_n}$ .

From the above, reach an estimate on the sum which is :

  $s_n+{\displaystyle \underset{n+1}{\overset{\infty }{\int }}{a}_n}\le R_n\le {\displaystyle \underset{n}{\overset{\infty }{\int }}{a}_n}.$

b.

To determine

To find: if a series is convergent by the Comparison Test, how do estimate its sum.

Answer to Problem 7RCC

  $s\le b_n-Q_n$ .

Explanation of Solution

Given information:

Calculation:

Recall the Comparison test for Convergence:

If have the series ${\displaystyle \sum a_n}\text{and}{\displaystyle {\sum }_n}$ .

If ${\displaystyle {\sum }_n}$ converges and:

  $a_n\le b_n\text{for}\text{all}\text{values}\text{of}n.$

Then ${\displaystyle {\sum }_n}$ also converges.

It follows simple logic then, that:

  ${\displaystyle \sum a_n}<{\displaystyle \sum b_n}$

Now the reminder of ${\displaystyle \sum a_n}$ is:

  $R_n=s_a-s_n.$

And the reminder of ${\displaystyle \sum b_n}$ is:

  $Q_n=q-q_n=1-(b_1+b_2+\mathrm{.....})$ .

Same logic shown above gives:

  $\begin{array}{c}{R}_n\le Q_n\\ s\le b_n-Q_n.\end{array}$

c.

To determine

To find: if a series is convergent by the Alternative Series Test, how do estimate its sum.

Answer to Problem 7RCC

  $\left|s-s_n\right|\le b_{n+1}.$

Explanation of Solution

Given information:

Calculation:

Recall the integration test for convergence:

If have the series:

  ${\displaystyle \sum _{n=1}^{\infty }{(-1)}^{n-1}}{b}_n.$

If:

  $\begin{array}{l}{b}_{n+1}\le b_n\\ \underset{n\to \infty }{\lim }{b}_n=0\end{array}$ .

The series Converges:

The Alternate Series Estimation Theorem states for the sum:

  $\left|s-s_n\right|\le b_{n+1}.$

Where $b_{n+1}$ is the first neglected term.