$\sum_{n=0}^\infty C_n(x-3)^n$ converges for $x=0$, and diverges for $x=7$. Does $\sum C_n$ converge? $\sum C_n5^n$? $\sum C_n\frac{2^{n+1}}{n+1}$?

Question

Suppose a power series $\sum_{n=0}^\infty C_n(x-3)^n$ converges when $x=0$ and diverges when $x=7$. Determine which series below will definitely converge.

I) $\sum_{n=0}^\infty C_n$

II) $\sum_{n=0}^\infty C_n5^n$

III) $\sum_{n=0}^\infty C_n\frac{2^{n+1}}{n+1}$

IV) $\sum_{n=0}^\infty C_n3^n$

Answer 1

Since that power series is centered at $3$, diverges at $7$ and converges at $0$, the radius of convergence is between $3$ and $4$. So, the series that will definitely converge are the series I) and III).