To determine
To calculate: The first 10 partial sums terms of the series and plot the sequence of terms and the sequence of partial sums on the graph to obtain the sum if the series convergent.
Answer
The first 10 partial sums of the series are 0.44721, 1.15432, 1.98637, 2.88080, 3.80927, 4.75796, 5.71948, 6.68962, 7.66581 and 8.64639.
The graph of the sequence of terms and the sequence of partial sums is shown below in Figure 1.
The series is divergent.
Explanation
Given:
The series is
∑n=1∞nn2+4
.
Here, the sequence of the term is
an=nn2+4
.
Result used:
The sum of the geometric series is
∑n=1∞arn−1=a1−r if r<1
.
Calculation:
The nth term of the partial sum is
sn=∑i=1nai
.
Obtain the first 10 terms of the sequence and partial sums.
n 
n2+4

an=nn2+4

sn=∑i=1nai

1 
12+4=2.236068

a1=0.447214

s1=0.447214

2 
22+4=2.828427

a2=0.707107

s2=1.154321

3 
32+4=3.605551

a3=0.832050

s3=1.986371

4 
42+4=4.472136

a4=0.894427

s4=2.880798

5 
52+4=5.385165

a5=0.928477

s5=3.809275

6 
62+4=6.324555

a6=0.948683

s6=4.757958

7 
72+4=7.280110

a7=0.961524

s7=5.719482

8 
82+4=8.246211

a8=0.970142

s8=6.689624

9 
92+4=9.219544

a9=0.976187

s9=7.665811

10 
102+4=10.198039

a10=0.980581

s10=8.646392

Therefore, the first 10 terms of the sequence are 0.44721, 0.70711, 0.83205, 0.89443,0.92848, 0.94868, 0.96152, 0.97014, 0.97619 and 0.98058
And the first 10 partial sums of the series are 0.44721, 1.15432, 1.98637, 2.88080, 3.80927, 4.75796, 5.71948, 6.68962, 7.66581, and 8.64639.
The graph of the sequence of terms and the sequence of partial sums is shown below in Figure 1.
From the graph, it is observed that the plotted points are not closer to zero.
Therefore, the series is divergent.