Question

Suppose the power series $\mathbf{\sum }_{\mathbf{k}\mathbf{=}\mathbf{0}}^{\mathbf{\infty }}{\mathbf{c}}_{\mathbf{k}}\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{4}{\mathbf{)}}^{\mathbf{K}}$is known to converge at -2and diverge at 13 . Discuss whether the series converges at -7,0,7,10, and 11. Possible answers are does, does not, might.

Short answer

Answer

The series does converges at 0 and 7 ; it does not converges at -7; and it might converges at 10 and 11.

Step-by-step solution

To Find the differential equation

The power series $\sum _{\mathrm{k}=0}^{\infty }{\mathrm{c}}_{\mathrm{k}}(\mathrm{x}-4{)}^{\mathrm{K}}$ converges at -2 and diverges at 13 . The center of the interval of convergence is a=4.

Since the series converges at -2, the interval is [-2,10) subset of the interval of convergence.

Since the series diverges at 13 , the interval of convergence is subset of the interval [-5,13) .

Now, we can conclude that the series does converges at 0 and 7 ; it does not converges at -7 ; and it might converges at 10 and 11.

Final Answer

The series does converges at 0 and 7 ; it does not converges at -7; and it might converges at 10 and 11.