Find the radius of convergence of $\sum_{n=1}^\infty n!(2x-1)^n$

Question

Now, by D'Alemberts Ratio test that implies (for convergence):

$\lim_{n\to\infty} \lvert \frac{(n+1)!(2x-1)(2x-1)^n}{n!(2x-1)^n} \rvert<1$

$\lim_{n\to\infty} \lvert (n+1)(2x-1)\rvert<1 $

$\lvert 2x-1\rvert<\lim_{n\to\infty} \lvert\frac{1}{(n+1)}\rvert$

$\lvert 2x-1\rvert < 0$

$\lvert 2x\rvert< 1$

$\frac{-1}{2}<x<\frac{1}{2}$

Therefore radius of convergence $(R) = \frac{1}{2}$

However upon inspection this answer is incorrect since for a value of $x = 0$, the sequence clearly diverges.

Can someone tell me where I have gone wrong?

Thanks a million!

MrYouMath I am unsure what you mean. And Henry is there a way to prove it using D'Alemberts ratio test? — user2250537, Apr 27 '16 at 20:42
The limit is infinity, so the radius of convergence is 0. That is, it will only converge for $x=\frac{1}{2}$ — Dave, Apr 27 '16 at 20:42
You cannot put $n+1$ in the denominator without putting the limit there as well. — Dave, Apr 27 '16 at 20:44
You've fixed your error now... you need $\lim_{n\rightarrow\infty}(1/(n+1)) > |2x-1|$. Since the left-hand limit is zero, the inequality is never satisfied, and so the radius of convergence is zero. — mjqxxxx, Apr 27 '16 at 20:55

MrYouMath · Accepted Answer · 2016-04-27T20:57:20.483

1

Assuming you mean

$$\sum_{n=1}^\infty n!(2x-1)^n=\sum_{n=1}^\infty n!2^n(x-0.5)^n$$

Now use ratio test:

$$\frac{|a_{n+1}|}{|a_n|}=\frac{(n+1)!2^{n+1}}{n!2^n}=2(n+1)>1$$ Inequality holds for every $n$. Hence the radius of convergence is $0$.

Formally you have to calculate

$$r=\lim_{n\to \infty}\frac{1}{\frac{|a_{n+1}|}{|a_n|}}=\lim_{n\to \infty}\frac{1}{2(n+1)}=0$$

Edit: Your method is wrong, because: $$\lvert 2x-1\rvert < 0$$ This inequality is can never be true for any $x$ as the absolute value is never negative.

edited Apr 27 '16 at 20:57

answered Apr 27 '16 at 20:45

MrYouMath

15,833

Is it not a power series though? How would you incorporate that element into the limit? – user2250537 Apr 27 '16 at 20:47
The ratio test for a power series only incorporates the coefficients. – MrYouMath Apr 27 '16 at 20:48
But what is wrong with my method though? – user2250537 Apr 27 '16 at 20:50
I still dont understant how that shows that the power series diverges – user2250537 Apr 27 '16 at 20:52
If your radius of convergence is $0$, this implies that your series only converges for $x=0.5$. It will diverge for all other values of $x$. – MrYouMath Apr 27 '16 at 20:54
MrYouMath, are you sure the series converges at x = 0.5? – user2250537 Apr 27 '16 at 21:00
$2x-1=2*0.5-1=0$. The sum will have the value $0$. Just check the sum. – MrYouMath Apr 27 '16 at 21:01
Oh ye thanks sorry I did something stupid – user2250537 Apr 27 '16 at 21:05

Find the radius of convergence of $\sum_{n=1}^\infty n!(2x-1)^n$

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