Can the constant term of a power series be zero?

[gikiian]

In the context of my work (linear differential equations), it can not be zero. But why?

 

[SteamKing]

In the context of my work (linear differential equations), it can not be zero. But why?

It's not clear why your context implies that the constant term cannot equal zero. If the constant term were zero, would the series not satisfy the differential equation and any associated initial conditions?

 

[D H]

y'' + y = 0, y(0) = 0, y'(0) = 1.

 

[gikiian]

I get it! It will satisfy the equation, but the solution will be a trivial one.

 

[D H]

There is only one solution for the initial value problem I wrote in post #3, and it is not the trivial solution.

 

[Mandelbroth]

In the context of my work (linear differential equations), it can not be zero. But why?

What about, say, a function $f:\mathbb{R}\to\mathbb{R}, x\mapsto x$?

 

[gikiian]

Can the constant term of a power series be zero?

Yes. Take $xe^{x}$ for example:
$xe^{x}$
$=x(1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\frac{x^{4}}{4!}+...)$
$=x+x^{2}+\frac{x^{3}}{2!}+\frac{x^{4}}{3!}+\frac{x^{5}}{4!}+...$
$=0+x+x^{2}+\frac{x^{3}}{2!}+\frac{x^{4}}{3!}+\frac{x^{5}}{4!}+...$

In the context of my work (linear differential equations), it can not be zero. But why?

In Frobenius method, you have $y=\sum^{∞}_{n=0} a_{n}x^{n+r}=x^{r}(a_{o}+a_{1}x+a_{2}x^{2}+...)$. Now if $a_{o}$ becomes 0, then the series would become $y=x^{r}(0+a_{1}x+a_{2}x^{2}+...)=x^{r}(a_{1}x+a_{2}x^{2}+...)=x^{r}x(a_{1}+a_{2}x+a_{3}x^{2}+...)=x^{r+1}(a_{1}+a_{2}x+a_{3}x^{2}+...)$. This would essentially change the mathematical technique that we use here to solve the ODE. Hence we 'assume' that $a_{o}$ can not be 0.

 

[AlephZero]

OK, now we can see this is a question about semantics, not series solutions of differential equations.

"The constant term of a power series" means $c_0$ in the series $c_0 + c_1x + c_2x^2 + \dots$.

But in post #7, you are just saying that every non-trivial series must have first non-zero term, and you are calling that term $a_0$. it is the coefficient of $x^r$. It is not the constant term of the series solution unless $r = 0$.

 

[gikiian]

OK, now we can see this is a question about semantics, not series solutions of differential equations.

"The constant term of a power series" means $c_0$ in the series $c_0 + c_1x + c_2x^2 + \dots$.

But in post #7, you are just saying that every non-trivial series must have first non-zero term, and you are calling that term $a_0$. it is the coefficient of $x^r$. It is not the constant term of the series solution unless $r = 0$.

I am a little confused as to what you are saying here, and would like to understand your point better. Please explain what are you trying to convey.