   Chapter 11.2, Problem 14E

Chapter
Section
Textbook Problem

# Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the 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.14. ∑ n = 1 ∞ ( sin 1 n − sin 1 n + 1 )

To determine

To calculate: The first 10 partial sums terms of the series and plot the sequence of terms and the sequence of partial sums on the graph to obtain the sum if the series convergent.

Explanation

Given:

The series is n=1(sin1nsin1n+1) .

Here, the sequence of the term is an=sin1nsin1n+1 .

Calculation:

The nth term of the partial sum is sn=i=1nai .

Obtain the first 10 terms of the sequence and partial sums.

 n an=sin1n−sin1n+1 sn=∑i=2nai 1 a1=0.36204545 s1=0.36204545 2 a2=0.15223084 s2=0.51427629 3 a3=0.07979074 s3=0.59406703 4 a4=0.04873463 s4=0.64280166 5 a5=0.03277312 s5=0.67557478 6 a6=0.02352440 s6=0.69909918 7 a7=0.01769700 s7=0.71679618 8 a8=0.01379210 s8=0.73058828 9 a9=0.01104921 s9=0.74163749 10 a10=0.00904949 s10=0.75068698

Therefore, the first 10 terms of the sequence are 0.36205, 0.15223, 0.07979, 0.04873, 0.03277, 0.02352, 0.01770, 0.01379, 0.01101 and 0.00905.

And the first 10 partial sums of the series 0.36205, 0.51428, 0.59407, 0.64280, 0.67557, 0.69910, 0.71680, 0.73059, 0.74164 and 0.75069.

The graph of the sequence of terms and the sequence of partial sums is shown below in Figure 1.

From the graph and the table, it is observed that the plotted points of the partial sums are closer to 1 and the terms of the sequence are closer to zero.

Therefore, the series is convergent.

Obtain the sum of the series.

That is, to compute the values of n=1an=n=1(sin1nsin1n+1)

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