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Sagot :
The power series representation for the function, so the interval of convergence is (-9,9).
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We have given f(x)=1/(9+x)
f(x)=(1/9)*(1/(1-(-x/9))
f(x)=summation of (n=0 to infinity) [1/9*(-x/9)n]
=summation of (n=0 to infinity) [(-1)n*(xn/9n+1)]
f(x)=summation of (n=0 to infinity) [(-1)n*(xn/9n+1)]
Let an=(-1)n*(xn/9n+1)
using the Ratio Test
L=lim n-->infinity|an+1/anL=lim n-->infinity|[(-1)n+1*(xn+1/9n+2)]/[(-1)n*(xn/9n+1)]|
=lim n-->infinity|[(-1)n+1*(xn+1/9n+2)]*[9n+1/(-1)n*(xn)]|
=lim n-->infinity|(-1)*(x)/9)|
=|-x/9|<1
x/9<1 and x>9>-1
x<9 and x>-9
x is -9<x<9
for x=9 the series ,summation of (n=0 to infinity) [(-1)n*(9n/9n+1)] =summation of (n=0 to infinity) [(-1)n*(/9)]
=1/9*summation of (n=0 to infinity) [(-1)n]
By the geometric series this summation of (n=0 to infinity) [(-1)n] diverges
=1/9*diverges
the series diverges for x=9
for x=-9 the series ,summation of (n=0 to infinity) [(-1)n*(-9)n/9n+1)] =summation of (n=0 to infinity) [(-1)2n*(/9)]
=1/9*summation of (n=0 to infinity) [(-1)2n]
which is diverges
so the interval of convergence is (-9,9)
Learn more about convergence here https://brainly.com/question/21089324
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