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.
Answer
The first 10 partial sums of the series are 0.54030, 0.12416, −0.86584, −1.51948, −1.23582, −0.27565, 0.47825, 0.33275, −0.57838 and −1.41745.
The graph of the sequence of terms and the sequence of partial sums is shown below in Figure 1.
The series is divergent.
Explanation
Given:
The series is
∑n=1∞cosn
.
Here, the sequence of the term is
an=cosn
.
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=cosn

sn=∑i=1nai

1 
a1=0.5403023

s1=0.5403023

2 
a2=−0.4161468

s2=0.1241555

3 
a3=−0.9899925

s3=−0.8658370

4 
a4=−0.6536436

s4=−1.5194806

5 
a5=0.2836622

s5=−1.2358184

6 
a6=0.9601703

s6=−0.2756481

7 
a7=0.7539023

s7=0.4782542

8 
a8=−0.1455000

s8=0.3327542

9 
a9=−0.9111303

s9=−0.5783761

10 
a10=−0.8390715

s10=−1.4174476

Therefore, the first 10 terms of the sequence are 0.54030, −0.41615, −0.98999, −0.65364, 0.28366, 0.96017, 0.75390, −0.14550, −0.91113 and −0.83907.
And the first 10 partial sums of the series are 0.54030, 0.12416, −0.86584, −1.51948, −1.23582, −0.27565, 0.47825, 0.33275, −0.57838 and −1.41745.
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 partial sums are not closer to zero.
Therefore, the series is divergent.