Respuesta :
Answer:
Converges for x in (-16,2) and the value of convergence is [tex]\frac{-7}{x+2}[/tex]
Step-by-step explanation:
Recall that a series of the form [tex]\sum_{n=0}^\infty r^n[/tex] converges if |r|<1. The value of convergence, when the series converges is
[tex]\frac{1}{1-r}[/tex]
The given series is [tex] \sum_{n=0}^\infty \frac{(x+9)^n}{7^n} = \sum_{n=0}^\infty(\frac{x+9}{7})^n[/tex]
So, it converges if and only if
[tex]\left|\frac{x+9}{7}\right|<1[/tex]
Equivalently, we get
[tex]|x+9|<7[/tex]
Which is the interval [tex]-16<x<-2[/tex]. So the series converges over the interval (-16,-2).
The value of converge for an x in that interval is
[tex] \frac{1}{1-\frac{x+9}{7}} = \frac{-7}{x+2}[/tex].
The value of x is [tex]-16 < x < -2[/tex] , for which the series converges.
The sum of the series is , [tex]\sum_{n=0}^{\infty }(\frac{x+9}{7} )^{n}[/tex]
Sum of series :
A series [tex]\sum_{n=0}^{\infty }(r^{n} )[/tex] is converges . if, [tex]|r| < 1[/tex]
Given series is, [tex]\sum_{n=0}^{\infty }(\frac{x+9}{7} )^{n}[/tex] converges, when :
[tex]|\frac{x+9}{7}| < 1\\ \\|x+9| < 7\\\\-16 < x < -2[/tex]
Thus, The value of x is [tex]-16 < x < -2[/tex] , for which the series converges.
Learn more about the convergent series here:
https://brainly.com/question/337693
