009C Sample Midterm 2, Problem 5

If  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_nx^n}   converges, does it follow that the following series converges?

(a)  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\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:
Assume that the power series  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_nx^n}   converges.
Let  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 R}   be the radius of convergence of this power series.
So, the power series
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_nx^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  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (-R,R),}
$\sum _{n=0}^{\infty }c_{n}{\bigg (}{\frac {a}{2}}{\bigg )}^{n}$  converges.
Since  $a$  was an arbitrary number in the interval  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (-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  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a\in (-R,R).}   Then,  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle -a\in (-R,R).}
Since  $\sum _{n=0}^{\infty }c_{n}x^{n}$  converges in the interval  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (-R,R),}
$\sum _{n=0}^{\infty }c_{n}(-a)^{n}$  converges.
Since  $a$  was an arbitrary number in the interval  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (-R,R),}
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}
converges in the interval  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 (-R,R).}

Final Answer:
(a)     converges
(b)     converges

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