Respuesta :
The series converges, and its sum is 1/2.
If r > 1, the series is divergent. If r < 1, the series is convergent. In our sequence, r, the common ratio we multiply by to get the next term, is 7/9; therefore it is convergent.
To find the sum of a convergent series, we use the formula
a/(1-r), where a is the first term and r is the common ratio. We then have
1/9÷(1-7/9) = 1/9÷2/9 = 1/9×9/2 = 1/2
If r > 1, the series is divergent. If r < 1, the series is convergent. In our sequence, r, the common ratio we multiply by to get the next term, is 7/9; therefore it is convergent.
To find the sum of a convergent series, we use the formula
a/(1-r), where a is the first term and r is the common ratio. We then have
1/9÷(1-7/9) = 1/9÷2/9 = 1/9×9/2 = 1/2
Answer:
In the given geometric series, r, the common ratio is 7/9; therefore the series is convergent.
Step-by-step explanation:
We are given the geometric series:
[tex]\frac{1}{9} + \frac{7}{81} + \frac{49}{729} + \frac{343}{6561} + ...[/tex]
The general geometric series is in the for [tex]a, ar, ar^2, ar^3, ...[/tex], where a is the first term and r is the common ratio.
Comparing with the given geometric series, we have, [tex]a = \frac{1}{9} \text{ and }r = \frac{7}{9}[/tex]
If r > 1, the series is divergent.
If r < 1, the series is convergent.
In the given geometric series, r, the common ratio is 7/9; therefore the series is convergent.
