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 −2.40000, −1.92000, −2.01600, −1.99680, −2.00064, −1.99987, −2.00003, −1.99999, −2.00000 and −2.00000.
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 −2.
Explanation
Given:
The series is
∑n=1∞12(−5)n
.
Here, the sequence of the term is
an=12(−5)n
.
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 
an=12(−5)n

sn=∑i=1nai

1 
a1=−2.4

s1=−2.4

2 
a2=0.48

s2=−1.92

3 
a3=−0.096

s3=−2.016

4 
a4=0.0192

s4=−1.9968

5 
a5=−0.00384

s5=−2.00064

6 
a6=0.000768

s6=−1.999872

7 
a7=−0.0001536

s7=−2.0000256

8 
a8=0.00003072

s8=−1.99999488

9 
a9=−0.000006144

s9=−2.000001024

10 
a10=0.0000012288

s10=−1.9999997952

Therefore, the first 10 terms of the sequence are −2.40000, 0.48000, −0.09600, 0.01920, −0.00384, 0.00077, −0.00015, 0.00003, 0.00000 and 0.00000.
And the first 10 partial sums of the series are −2.40000, −1.92000, −2.01600, −1.99680, −2.00064, −1.99987, −2.00003, −1.99999, −2.00000 and −2.00000.
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 partial sums are closer to −2.
Therefore, the series is convergent.
Obtain the sum of the series.
That is, to compute the values of
∑n=1∞an=∑n=1∞12(−5)n
.
∑n=1∞12(−5)n=∑n=1∞12⋅1(−5)n=∑n=1∞12⋅(−15)n
Clearly, it is geometric series with
a=−125 and r=−15
.
Since
−1<−15<1
and by the result stated above,
∑n=1∞12(−5)n=−1251−(−15)=−1251+15=−12565=−2
Thus, the sum of the series is −2.
Therefore, the series convergent to the sum is −2.