Get the solution to "Given the series: 9+1174+152116+1977364+... does this series converge..."

postillan4

Answered question

2021-02-03

Given the series:

9 + \frac{117}{4} + \frac{1521}{16} + \frac{19773}{64} + . . .

does this series converge or diverge? If the series converges, find the sum of the series.

Answer & Explanation

dieseisB

Skilled2021-02-04Added 85 answers

To determine
Given:
An infinite series $9 + \frac{117}{4} + \frac{1521}{16} + \frac{19773}{64} + . . .$
To determine:
Given series is convergent or divergent.
Calculation
Consider the given series:
$9 + \frac{117}{4} + \frac{1521}{16} + \frac{19773}{64} + . . .$
Here, $a_{1} = 9$
$a_{2} = \frac{117}{4}$
$a_{3} = \frac{1521}{16}$ and so on.
Now, $\frac{a_{2}}{a_{1}} = \frac{\frac{117}{4}}{9} = \frac{13}{4}$
Similarly, $\frac{a_{3}}{a_{2}} = \frac{\frac{1521}{16}}{\frac{117}{4}} = \frac{13}{4}$
Similarly, $\frac{a_{4}}{a_{3}} = \frac{\frac{19773}{64}}{\frac{1521}{16}} = \frac{13}{4}$
Here, $\frac{a_{2}}{a_{1}} = \frac{a_{3}}{a_{2}} = \frac{a_{4}}{a_{3}}$
Therefore, given series is geometric series with common ratio $r = \frac{13}{4}$
Here, $r = \frac{13}{4} > 1$
We know, a geometric series with common ratio $> 1$ is always divergent.
Therefore, the given series is divergent.

Jeffrey Jordon

Expert2021-12-27Added 2605 answers

Answer is given below (on video)

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