-z.54 find the radius of convergence

In summary, the radius of convergence for the given power series is $1/6$ and this can be obtained by examining the coefficient of the first term and taking the limit of the $n$-th root of the coefficient.
  • #1
karush
Gold Member
MHB
3,269
5
$\tiny{10.7.37}$
$\displaystyle\sum_{n=1}^{\infty}
\frac{6\cdot 12 \cdot 18 \cdots 6n}{n!} x^n$
find the radius of convergence
I put 6 but that wasn't the answer
 
Last edited:
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  • #2
Please explain how you originally obtained $6$ as the answer.
 
  • #3
the ans was 1/6
Looked at an example very close to this
and noticed the first term revealed the answer but couldn't follow all the steps they had to get it.
 
  • #4
Well, the coefficient $a_n$ of $x^n$ in the power series reduces to $6^n$, for $6\cdot 12\cdot 18\cdots 6n = (6\cdot 1)(6\cdot 2)\cdots (6\cdot n) = 6^nn!$. So, $\sqrt[n]{a_n} = 6$, and the radius $R$ of convergence of the power series is given by $1/\lim\limits_{n\to \infty} \sqrt[n]{a_n} = 1/6$.
 

FAQ: -z.54 find the radius of convergence

What is the meaning of "radius of convergence" in -z.54 find the radius of convergence?

The radius of convergence refers to the distance from the center of a power series to the nearest point where the series converges. In the context of -z.54 find the radius of convergence, it is the distance from the center of the power series -z.54 to the nearest point where the series converges.

How is the radius of convergence calculated?

The radius of convergence can be calculated using the ratio test or the root test. These tests involve taking the limit of the absolute value of the ratio or root of the coefficients of the power series. If the limit is less than 1, the series converges, and the radius of convergence can be found by taking the reciprocal of the limit.

Can the radius of convergence be negative?

No, the radius of convergence cannot be negative. It represents a distance and must be a positive value.

What does it mean if the radius of convergence is infinite?

If the radius of convergence is infinite, it means that the power series converges for all values of the variable. This is also known as a "power series with infinite radius of convergence."

Why is it important to find the radius of convergence?

Knowing the radius of convergence is important because it tells us for which values of the variable the power series will converge. This allows us to determine the domain of convergence and to use the power series to approximate functions within that domain.

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