The radius of convergence of given power series is [tex]\frac{1}{\rho} = \limsup_n \sqrt[n^2]{|c_n|}[/tex].
According to the statement
we have to find that the radius of convergence of the given power series.
So, For this purpose, we know that the
The radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges.
So,
If R is the radius of convergence of [tex]\sum_{n=0}^{\infty}c_n x^n[/tex] then the series converges absolutely if |x|<R and diverges if |x|>R.
Hence, [tex]\sum_{n=0}^{\infty}c_n x^{2n}= \sum_{n=0}^{\infty}c_n (x^2)^n[/tex] converges absolutely if |x^2|<R, and diverges if |x^2|>R. Hence the radius of convergence must be [tex]\sqrt{R}[/tex].
For the second, note that if we write the power series as
[tex]\sum_{n=0}^{\infty}a_n x^n[/tex] then an=0 if n is not a square, and [tex]a_n =c_k[/tex] if [tex]n=k^2[/tex].
Hence the radius of convergence is [tex]\frac{1}{\rho} = \limsup_n \sqrt[n^2]{|c_n|}[/tex].
So, The radius of convergence of given power series is [tex]\frac{1}{\rho} = \limsup_n \sqrt[n^2]{|c_n|}[/tex].
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