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 0.36205, 0.51428, 0.59407, 0.64280, 0.67557, 0.69910, 0.71680, 0.73059, 0.73059, 0.74164 and 0.75069.
The graph of the sequence of terms and the sequence of partial sums is shown below in Figure 1.
The series is convergent to the sum 0.84147.
Explanation
Given:
The series is
∑n=1∞(sin1n−sin1n+1)
.
Here, the sequence of the term is
an=sin1n−sin1n+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 
an=sin1n−sin1n+1

sn=∑i=2nai

1 
a1=0.36204545

s1=0.36204545

2 
a2=0.15223084

s2=0.51427629

3 
a3=0.07979074

s3=0.59406703

4 
a4=0.04873463

s4=0.64280166

5 
a5=0.03277312

s5=0.67557478

6 
a6=0.02352440

s6=0.69909918

7 
a7=0.01769700

s7=0.71679618

8 
a8=0.01379210

s8=0.73058828

9 
a9=0.01104921

s9=0.74163749

10 
a10=0.00904949

s10=0.75068698

Therefore, the first 10 terms of the sequence are 0.36205, 0.15223, 0.07979, 0.04873, 0.03277, 0.02352, 0.01770, 0.01379, 0.01101 and 0.00905.
And the first 10 partial sums of the series 0.36205, 0.51428, 0.59407, 0.64280, 0.67557, 0.69910, 0.71680, 0.73059, 0.74164 and 0.75069.
The graph of the sequence of terms and the sequence of partial sums is shown below in Figure 1.
From the graph and the table, it is observed that the plotted points of the partial sums are closer to 1 and the terms of the sequence are closer to zero.
Therefore, the series is convergent.
Obtain the sum of the series.
That is, to compute the values of
∑n=1∞an=∑n=1∞(sin1n−sin1n+1)
.
∑n=1∞(sin1n−sin1n+1)=limn→∞(∑k=1n(sin1k−sin1k+1))
(1)
Obtain the values of the series
∑k=1n(sin1k−sin1k+1)
.
∑k=1n(sin1k−sin1k+1)=(sin11−sin11+1)+(sin12−sin12+1)+⋯+(sin1n−sin1n+1)=(sin11−sin12)+(sin12−sin13)+⋯+(sin1n−sin1n+1)=sin11−sin12+sin12−sin13+⋯+sin1n−sin1n+1
∑k=1n(sin1k−sin1k+1)=sin1−sin1n+1
(2)
Substitute equation (2) in equation (1),
∑n=1∞(sin1n−sin1n+1)=limn→∞(sin1−sin1n+1)=limn→∞(sin1)−limn→∞(sin1n+1)=sin1−sin(0)=sin1(≈0.84147)
Thus, the series sum is
0.84147
.
Therefore, the series convergent to the sum is
0.84147
.