   Chapter 11.2, Problem 12E

Chapter
Section
Textbook Problem

Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why. ∑ n = 1 ∞ 7 n + 1 10 n

To determine

To find:

At least 10 partial sums of the series and graph both the sequence of terms and the sequence of partial sums on the same screen then conclude about the convergence or divergence of the series then find the sum.

Explanation

1) Concept:

Use a calculator to find at least 10 partial sums, sequence of terms and graph, then usethe geometric series to find the sum.

2) Geometric series:

n=1arn-1=a+ar+ar2+

is convergent, if r<1 and its sum is

n=1arn-1=a1-rr<1

If r1, the geometric series is divergent

3) Given:

n=17n+110n

4) Calculation:

The table shows 10 partial sums of the series sn and the sequence of terms  an:

 n an sn 1 4.90000 4.90000 2 3.43 8.33000 3 2.401 10.73100 4 1.6807 12.41170 5 1.17649 13.58819 6 0.82354 14.41173 7 0.57648 14.98821 8 0.403536 15.39175 9 0.28248 15.67422 10 0.19773 15.87196

Graph:

From the above graph and table, it can be observe that the terms of the sequence are getting smaller and may approach 0 and the partial sum converges to a value near 16.

Use following steps for TI-86 Calculator:

To graph {an} and  {sn}, set the calculator to Param mode in under mode button and DrawDot mode.

(DrawDot is under GRAPH, MORE, FORMT (F3).)

Now, under Et=  make the assignments:

xt1=t, yt1=7t+110t, xt2=t, yt2=sum seqyt1,t,1,t,1

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