Question

12. [12 points total] Consider the following power series: $$sum_{k=1}^{infty} frac{2^k(x-3)^k}{k}$$. Use the Ratio Test to find the radius of convergence. Then determine the interval of convergence, checking the endpoints if necessary. Write your answers in the spaces provided below. You MUST show your work and explain your reasoning! Hint: You may want to use the fact that if $b > 1$ then $lim_{k oinfty} frac{b^k}{k} eq 0$. 13. [12 points total] We know that we can write $frac{1}{1-x} = sum_{k=0}^{infty} x^k$ for $-1 < x < 1$. Follow the prescribed steps to manipulate this power series to get a power series representation for $f(x) = an^{-1}(x)$ centered at $a = 0$. • Make the appropriate substitution to find a power series for $g(x) = 1/(1 + x^2)$ centered at $a = 0$. You may do this part in multiple steps if you wish. • Integrate the previous power series to find a power series for $f(x) = an^{-1}(x)$ centered at $a = 0$. Has the radius of convergence changed? Has the interval of convergence changed?

          12. [12 points total] Consider the following power series:
$$sum_{k=1}^{infty} frac{2^k(x-3)^k}{k}$$.
Use the Ratio Test to find the radius of convergence. Then determine the interval of convergence, checking the endpoints if necessary. Write your answers in the spaces provided below. You MUST show your work and explain your reasoning!
Hint: You may want to use the fact that if $b > 1$ then $lim_{k	oinfty} frac{b^k}{k} 
eq 0$.
13. [12 points total] We know that we can write $frac{1}{1-x} = sum_{k=0}^{infty} x^k$ for $-1 < x < 1$. Follow the prescribed steps to manipulate this power series to get a power series representation for $f(x) = 	an^{-1}(x)$ centered at $a = 0$.
• Make the appropriate substitution to find a power series for $g(x) = 1/(1 + x^2)$ centered at $a = 0$. You may do this part in multiple steps if you wish.
• Integrate the previous power series to find a power series for $f(x) = 	an^{-1}(x)$ centered at $a = 0$. Has the radius of convergence changed? Has the interval of convergence changed?

$12. [12 points total] Consider the following power series: sumk=1^infty frac2^k(x-3)^kk . Use the Ratio Test to find the radius of convergence. Then determine the interval of convergence, checking the endpoints if necessary. Write your answers in the spaces provided below. You MUST show your work and explain your reasoning! Hint: You may want to use the fact that if b > 1 then limk oinfty fracb^kk eq 0. 13. [12 points total] We know that we can write frac11-x = sumk=0^infty x^k for -1 < x < 1. Follow the prescribed steps to manipulate this power series to get a power series representation for f(x) = an^-1(x) centered at a = 0. • Make the appropriate substitution to find a power series for g(x) = 1/(1 + x^2) centered at a = 0. You may do this part in multiple steps if you wish. • Integrate the previous power series to find a power series for f(x) = an^-1(x) centered at a = 0. Has the radius of convergence changed? Has the interval of convergence changed?$

Added by Vicenta W.

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