To determine
To calculate: The first eight terms of the sequence of the partial sums and use them to check whether the series is convergent or divergent.
Answer
The first eight terms of the sequence of the partial sums are 0.8415, 1.7508, 1.8919, 1.1351, 0.1762, −0.1033, 0.5537, and 1.5431.
The series is divergent.
Explanation
Given:
The series is
∑n=1∞sinn
.
Here,
an=sinn
.
Calculation:
The nth term of the partial sum is
sn=∑i=1nai
.
Obtain the first eight terms of the sequence of the partial sums.
n 
an=sinn

sn=∑i=1nai

1 
a1=0.84147098

s1=0.84147098

2 
a2=0.90929743

s2=1.75076841

3 
a3=0.14112001

s3=1.89188842

4 
a4=−0.7568025

s4=1.13508592

5 
a5=−0.95892427

s5=0.17616165

6 
a6=−0.27941550

s6=−0.10325385

7 
a7=0.65698660

s7=0.55373275

8 
a8=0.98935825

s8=1.543091

Therefore, the first eight terms of the sequence of the partial sums are 0.8415, 1.7508, 1.8919, 1.1351, 0.1762, −0.1033, 0.5537, and 1.5431.
From the last column of the table, it is observed that the sequence of partial sums
{sn}
is divergent.
If the sequence of partial sums is divergent, then that series is divergent.
Therefore, the series is divergent.