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 1, 0.5, 0.6667, 0.6250, 0.6333, 0.6319, 0.6321, and 0.6321.
The series is convergent.
Explanation
Given:
The seriesis
∑n=1∞(−1)n−1n!
.
Here,
an=(−1)n−1n!
.
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 
1n!

an=(−1)n−1n!

sn=∑i=1nai

1 
11!=1

a1=1

s1=1

2 
12!=0.5

a2=−0.5

s2=1.5

3 
13!=0.16666667

a3=0.16666667

s3=0.66666667

4 
14!=0.0416667

a4=−0.0416667

s4=0.62499997

5 
15!=0.0083333

a5=0.0083333

s5=0.63333327

6 
16!=0.0013889

a6=−0.0013889

s6=0.63194437

7 
17!=0.0001984

a7=0.0001984

s7=0.63214277

8 
18!=0.0000248

a8=−0.0000248

s8=0.63211797

Therefore, the first eight terms of the sequence of the partial sums are 1, 0.5, 0.6667, 0.6250, 0.6333, 0.6319, 0.6321, and 0.6321.
From the last column of the table, it is observed that the sequence of partial sums
{sn}
is closer to 0.6 (approximately).
Thus, the sequence is convergent.
If the sequence of partial sums is convergent, then the series is convergent.
Therefore, the series is convergent.