009C Sample Midterm 2, Problem 5

If  $\sum _{n=0}^{\infty }c_{n}x^{n}$  converges, does it follow that the following series converges?

(a)  $\sum _{n=0}^{\infty }c_{n}{\bigg (}{\frac {x}{2}}{\bigg )}^{n}$

(b)  $\sum _{n=0}^{\infty }c_{n}(-x)^{n}$

Foundations:
If a power series converges, then it has a nonempty interval of convergence.

Solution:

(a)

Step 1:
Assume that the power series  $\sum _{n=0}^{\infty }c_{n}x^{n}$  converges.
Let  $R$  be the radius of convergence of this power series.
So, the power series
$\sum _{n=0}^{\infty }c_{n}x^{n}$
converges in the interval  $(-R,R).$

Step 2:
Let  $a\in (-2R,2R).$  Then,  ${\frac {a}{2}}\in (-R,R).$
Since  $\sum _{n=0}^{\infty }c_{n}x^{n}$  converges in the interval  $(-R,R),$
$\sum _{n=0}^{\infty }c_{n}{\bigg (}{\frac {a}{2}}{\bigg )}^{n}$  converges.
Since  $a$  was an arbitrary number in the interval  $(-2R,2R),$
$\sum _{n=0}^{\infty }c_{n}{\bigg (}{\frac {x}{2}}{\bigg )}^{n}$
converges in the interval  $(-2R,2R).$

(b)

Step 1:
Assume that the power series  $\sum _{n=0}^{\infty }c_{n}x^{n}$  converges.
Let  $R$  be the radius of convergence of this power series.
So, the power series
$\sum _{n=0}^{\infty }c_{n}x^{n}$
converges in the interval  $(-R,R).$

Step 2:
Let  $a\in (-R,R).$  Then,  $-a\in (-R,R).$
Since  $\sum _{n=0}^{\infty }c_{n}x^{n}$  converges in the interval  $(-R,R),$
$\sum _{n=0}^{\infty }c_{n}(-a)^{n}$  converges.
Since  $a$  was an arbitrary number in the interval  $(-R,R),$
$\sum _{n=0}^{\infty }c_{n}(-x)^{n}$
converges in the interval  $(-R,R).$

Final Answer:
(a)     converges
(b)     converges

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