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, 1.7937, 2.4871, 3.1170, 3.7018, 4.2521, 4.7749, and 5.2749.
The series is divergent.
Explanation
Given:
The series is
∑n=1∞1n3
.
Here,
an=1n3
.
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=1n3

sn=∑i=1nai

1 
a1=1

s1=1

2 
a2=0.7937005

s2=1.7937005

3 
a3=0.6933613

s3=2.4870618

4 
a4=0.6299605

s4=3.1170223

5 
a5=0.5848035

s5=3.7018258

6 
a6=0.5503212

s6=4.2521470

7 
a7=0.5227580

s7=4.7749050

8 
a8=0.5000000

s8=5.2749050

Therefore, the first eight terms of the sequence of the partial sums are 1, 1.7937, 2.4871, 3.1170, 3.7018, 4.2521, 4.7749, and 5.2749.
From the last column of the table, it is observed that the sequence of partial sums
{sn}
is increasing. That is,
s1<s2<s3<s4<s5<s6<s7<s8
.
Thus, the sequence is divergent.
If the sequence of partial sums is divergent, then that series is divergent.
Therefore, the series is divergent.