# Thread: Root of a Power Series

1. ## Root of a Power Series

I have to find the "second smallest root" of the following equation :

$1-x+(x^2)/(2!)^2-(x^3)/(3!)^2+(x^4)/(4!)^2+...=0$

Matlab returns quite a satisfactory answer. >> p=[1/518400 -1/14400 1/1576 -1/36 1/4 -1 1]

p =
0.0000 -0.0001 0.0006 -0.0278 0.2500 -1.0000 1.0000
>> roots(p)
ans =
35.5690
-4.6796 +18.5352i
-4.6796 -18.5352i
4.1776 + 3.2154i
4.1776 - 3.2154i
1.4350

But I have been asked to identify this series as well, which I am unable to do. Can anybody help me identify this series as a function or a product of functions? Thanking anybody who answers before hand

2. Are you asking about roots for the infinite power series equation? Roots of the 6th degree polynomial formed by the first 7 terms do not tell you anything about roots of the entire series. This looks like the power series for $\displaystyle e^{-x}$ which is never 0.

3. Originally Posted by HallsofIvy
[snip]
This looks like the power series for $\displaystyle e^{-x}$ which is never 0.
The only problem with that observation being that the denominator of each coefficient has the form (n!)^2 rather than n! ....

4. AS Mr. Fanatstic pointed out the power series of e^x is 1+x+(x^2)/(2!)+(x^3)/(3!)+...

The problem with this series is the square of the nth factorial is involved.

5. Originally Posted by sparklingway
I have to find the "second smallest root" of the following equation :

$1-x+(x^2)/(2!)^2-(x^3)/(3!)^2+(x^4)/(4!)^2+...=0$

Matlab returns quite a satisfactory answer. >> p=[1/518400 -1/14400 1/1576 -1/36 1/4 -1 1]

p =
0.0000 -0.0001 0.0006 -0.0278 0.2500 -1.0000 1.0000
>> roots(p)
ans =
35.5690
-4.6796 +18.5352i
-4.6796 -18.5352i
4.1776 + 3.2154i
4.1776 - 3.2154i
1.4350

But I have been asked to identify this series as well, which I am unable to do. Can anybody help me identify this series as a function or a product of functions? Thanking anybody who answers before hand
I believe the function you seek is

$\displaystyle J_0(2\sqrt{x})$

where $\displaystyle J_0$ is a Bessel function of the first kind. If correct, the second smallist zero I got using Maple is

$\displaystyle x = 7.61781559$

6. Thanks. I got the problem solved yesterday