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)  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Step 2:

(b)

Step 1:

Step 2:

Final Answer:
(a)     converges
(b)     converges

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