Question

Suppose that P[infinity] n=0 cnx n converges when x = −5 and diverges when x = 7. What can be said about the convergence or divergence of the following series? (a) X[infinity] n=0 cn

Answer 1

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

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