Power series: x^3/3 + x^9/9 + x^15/15 .......

[Shubhan672]

Moved from technical forum section, so missing the homework template

Let x be a real number. Find the function whose power series is represented as follows: x^3/3! + x^9/9! + x^15/15! ...

I see that there is a connection to the power series expansion of e^x but am having difficulty finding the function.

 

[Mark44]

Moved from technical forum section, so missing the homework template
Let x be a real number. Find the function whose power series is represented as follows: x^3/3! + x^9/9! + x^15/15! ...

I see that there is a connection to the power series expansion of e^x but am having difficulty finding the function.

Notice that $x^3 = (x^3)^1$ and $x^9 = (x^3)^3$. Does that give you any ideas?

 

[vela]

Try differentiating the series multiple times and see if you can determine a differential equation it satisfies.

 

[Shubhan672]

Try differentiating the series multiple times and see if you can determine a differential equation it satisfies.

f(x) =

f'(x) =

f''(x) =

I can see that the 6 th, 12th, 18th ... derivative of f(x) would be f(x) itself but am having difficulty understanding its relevance and am having difficulty proceeding from here,

 

[PeroK]

f(x) = View attachment 256852

f'(x) = View attachment 256853

f''(x) = View attachment 256862

I can see that the 6 th, 12th, 18th ... derivative of f(x) would be f(x) itself but am having difficulty understanding its relevance and am having difficulty proceeding from here,

$f^{(6)}(x) = f(x)$ is a differential equation. Can you solve it?

Hint: complex roots of unity.