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 4.90000, 8.33000, 10.73100, 12.41170, 13.58819, 14.41173, 14.98821, 15.39175, 15.67422 and 15.87196.
Explanation
Given:
The series is ∑n=1∞7n+110n.
Here, the sequence of the term is an=7n+110n.
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=7n+110n | sn=∑i=1nai |
| 1 | a1=4.9 | s1=4.9 |
| 2 | a2=3.43 | s2=8.33 |
| 3 | a3=2.401 | s3=10.731 |
| 4 | a4=1.6807 | s4=12.4117 |
| 5 | a5=1.17649 | s5=13.58819 |
| 6 | a6=0.823543 | s6=14.411733 |
| 7 | a7=0.5764801 | s7=14.9882131 |
| 8 | a8=0.40353607 | s8=15.39174917 |
| 9 | a9=0.282475249 | s9=15.674224419 |
| 10 | a10=0.1977326743 | s10=15.8719570933 |
Therefore, the first 10 terms of the sequence are 4.90000, 3.43000, 2.40100, 1.68070, 1.17649, 0.82354, 0.57648, 0.40353, 0.28248 and 0.19773.
And the first 10 partial sums of the series 4.90000, 8.33000, 10.73100, 12.41170, 13.58819, 14.41173, 14.98821, 15.39175, 15.67422 and 15.87196.
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 16 when the terms of the sequence approach 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∞7n+110n.
∑n=1∞7n+110n=∑n=1∞7⋅7n10n=∑n=1∞7⋅(710)n=7⋅∑n=1∞(710)n
Since −1<710<1 and by the result stated above,
∑n=1∞7n+110n=7⋅7101−710=4910310=493=16.3333
Thus, the sum of the series is 16.3¯.
Therefore, the series is convergent to the sum is 16.3¯.