# 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.

Question
Series
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.

2021-02-04
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.

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