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.