What can be said about the convergence or divergence of the following series? summation n=0 to infinite (-1)^n Cn9^n
Answer:
The series diverges
Step-by-step explanation:
Given.
The center of convergence of the series, ΣCnx^n = 0 (for n = 0 to ∞)
The interval of the convergence is of the form (-k,k)
One or both the end points may be included in the intervals of convergence.
Also, given that the series diverges when x = 7; this implies that the interval of convergence is smaller than {-7,7}
The series
Σ(-1)^n Cn9^n (for n = 0 to ∞)
= Σ Cn(-9)^n (for n = 0 to ∞)
This is obtained after substituting -9 for x in ΣCnx^n = 0
So, x = -9
This is outside the boundary {-7,7}.
Hence, the given series diverges
