What is True about the radius of convergence R of a power series \sum_{n=1}^{\infty} a_n (x-a)^n ? Power series always converges for x=a , so let x
eq a . Taylor series T_{\infty}(x,a) for a function f is a power series with a_n=\frac{f^{(n)}(a)}{n!} .

Question

What is True about the radius of convergence   R  of a power series    \sum_{n=1}^{\infty} a_n (x-a)^n ? Power series always converges for   x=a , so let   x
eq a . Taylor series   T_{\infty}(x,a)  for a function   f  is a power series with   a_n=\frac{f^{(n)}(a)}{n!} .

Answer

\frac{1}{R}=\lim_{n	o \infty} \left | \frac{a_n}{a_{n+1}}ight |   when the limits exists.

Answer

If   a_n=\frac{1}{n!}  then   R=0 .

Answer

Taylor series for   f  and Taylor series for   \int_a^x f(t)dt  have the same   R .

Answer

Series is diverging for    |x-a|>R .

Answer

\frac{1}{R}=\lim_{n	o \infty} |a_n|^{\frac{1}{n}}  when the limits exists.

Answer

Series is absolutely converging for    |x-a| < R .

Answer

Series is uniformly converging for   |x-a|\leq \rho  where   \rho < R  .

Answer

If   a_n=n!  then   R=\infty .

Answer

Taylor series for   f  and Taylor series for   f'  have the same   R .

Answer

Series may or may not converge for    |x-a|=R .

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What is True about the radius of convergence R of a power series \sum_{n=1...

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