help with power series

[maeveoneill]

Just wondering if I am on the right track...?

Find the radius of convergence and interval of convergence of the series.

sum of the series from n=1 to infinity --> n!(x^n) / 1 . 3. 5 .... (2n-1)!

first of all is the equation the same as ---> n!(x^n) / (2n-1)!

if this is so... i have started to use the ratio test

lim ''''''''''''''''''''''''''''''''''''''''''' n+1!(x^(n+1)) ''''''''''''' (2n-1)! '''''''''' (n+1)x . (2n-1)!
n--> infinity ''''''''''''''''' ----------------- x ----------- = -------------------
'''''''''''''''''''''''''''''''''''''''''''''''''''''' (2n+1)! ''''''''''''''''''''''''''' n!(x^n) '''''''''''''''''' (2n+1)!

 

[pka]

\(\displaystyle \sum\limits_{n = 1}^\infty {\frac{{n!x^n }}{{1 \cdot 3 \cdots \cdot \left( {2n - 1} \right)}}}\)
I suspect that the above is correct. I think you have a missplaced "!".

 

[soroban]

Hello,maeveoneill!

Find the radius of convergence and interval of convergence of the series.

. . \(\displaystyle \sum^{\infty}_{n=1}\,\frac{n!x^n}{1\cdot3\cdot5\cdots(2n-1)}\)
Is the function the same as: .\(\displaystyle \frac{n!x^n}{(2n-1)!}\) . . . . no

\(\displaystyle \text{Ratio test: }\;R \;=\;\frac{(n+1)!x^{n+1}} {1\cdot3\cdot5\cdots(2n+1)} \cdot \frac{1\cdot3\cdot5\cdots(2n-1)}{n!x^n}\)

. . \(\displaystyle = \;\frac{(n+1)!}{n!}\cdot\frac{x^{n+1}}{x^n}\cdot\frac{1\cdot3\cdot5\cdots(2n-1)}{1\cdot3\cdot5\cdots(2n+1)} \;=\;\frac{(n+1)x}{2n+1}\)


\(\displaystyle \lim_{n\to\infty}\,\frac{(n+1)x}{2n+1} \;=\;\lim_{n\to\infty}\,\frac{\left(1+\frac{1}{n}\right)x}{2+\frac{1}{n}} \;=\;\frac{1}{2}x\)


. . \(\displaystyle \text{Then: }\;\left|\frac{x}{2}| \:< \:1\quad\Rightarrow\quad -1 \:<\: \frac{x}{2} \:<\: 1 \quad\Rightarrow\quad -2 \:<\: x \:<\:2\)


\(\displaystyle \text{Therefore: }\;\begin{array}{cc}\text{Radius of convergence:} & r \,=\,2 \\ \text{Interval of convergence: }& (-2,\,2)\end{array}\)

 

[pka]

Sometimes it just must be said.
Giving solutions to students allows them to get by homework.
Allowing them to find the solution insures that they understand.
Passing homework and passing tests are to different things.
At least, that is true for college level work.
I don’t know what goes at Junior Colleges.