How is $t^2$ a power series? With reference to following paragraph from Martin Braun, how is $t^2$ a power series? Is it because ratio is 0? Why is it permissible (It does not remains series anymore)?
 A: A power series is an infinite sum of the type $\sum_{n=0}^{\infty} a_n t^n$ for some (real? complex? depending on the context) $a_n$.
In the case where $a_n=0$ for all $n\not=2$ and $a_2=1$ you would get that $\sum_{n=0}^{\infty} a_n t^n=t^2$, and so it is a power series. Do note that this specific power series absolutely converges for all $t$.
A: $t^2$ is a power series $\sum_{n=0}^\infty c_n t^n$, where $c_2=1$ and $c_n=0$, for all $n \neq 2$.
A: A polynomial is a special case of a power series. But I found that the paragraph in the included page is confusing.
First, the power series for $y(t)$ above the green-highlighted text is clearly not a polynomial. So I cannot see why the author can compare with the power series $t^2$ and $2t$.
Second, the author mentioned the Cauchy ratio test. This test can be carried out with the power series for $y(t)$, but not for the power series $t^2$ and $2t$ because the latter have zero coefficients everywhere except one.
Thus, I think you should post the full page to provide more context to the paragraph in the book.
