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 0.5, 0.55, 0.5611, 0.5648, 0.5663, 0.5671, 0.5675, and 0.5677.
The series is convergent.
Explanation
Given:
The series is
∑n=1∞1n4+n2
.
Here,
an=1n4+n2
.
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 
n4+n2

an=1n4+n2

sn=∑i=1nai

1 
14+12=2

a1=0.5

s1=0.5

2 
24+22=20

a2=0.05

s2=0.55

3 
34+32=30

a3=0.0111

s3=0.5611

4 
44+42=272

a4=0.0037

s4=0.5648

5 
54+52=650

a5=0.0015

s5=0.5663

6 
64+62=1332

a6=0.0008

s6=0.5671

7 
74+72=2450

a7=0.0004

s7=0.5675

8 
84+82=4160

a8=0.0002

s8=0.5677

Therefore, the first eight terms of the sequence of the partial sums are 0.5, 0.55, 0.5611, 0.5648, 0.5663, 0.5671, 0.5675, and 0.5677.
From the last column of the table, it is observed that the sequence of partial sums
{sn}
is approximately closer to 0.6.
Thus, the sequence is convergent.
If the sequence of partial sum is convergent, then that series is convergent.
Therefore, the series is convergent.