Power series Question

Question

Suppose that 

∞

∑ c_n x ⁿ

n = 0

 converges when x = −4 and diverges when x = 6. What can be said about the convergence or divergence of the following series?

∞

∑ (−1)ⁿc_n7ⁿ

n = 0

When compared to the original series, we see that x =  ? here. Since the original series diverges for that particular value of x, we know that this series diverges.

(my answer for x was 7 but its not the correct answer so I was so confused what other value could x be?)

Answer 1

The given power series is centered at 0, converges at x = -4, and diverges at x = 6.

So, the interval of convergence has one of the following forms (-a,a), [-a,a], [-a,a). or (-a,a], where a is some number in the interval 4 ≤ a ≤ 6.

When x = -7, we have ∑_{(n=0 to ∞)} c_n(-7)ⁿ = ∑_{(n=0 to ∞)}c_n(-1)ⁿ7ⁿ.

Since -7 is not in the interval of convergence, the series diverges.

Answer 2

your x=(-)7

so it is an alternating series sum, and terms must approach zero

for series sum to converge.

if c_n=1/5ⁿ, series would converge for x=-4 but not x=--7, and diverge for x=6

if c_n=1/8ⁿ series would converge for x=-4 and x=-7 and converge for x=6, meaning c_n>1/8ⁿ

so x=6 converges.

based on these simple c_n, series will diverge when x==-7.