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A series with all its terms are functions defined on interval \(I\) called function series \(\displaystyle \sum_{n=1}^{\infty} u_n(x)\), for every given specific \(x_0\in I\), the function series constructs a constant term series \(\displaystyle \sum_{n=1}^{\infty} u_n(x_0)\). If the series converges at \(x_0\), we name \(x_0\) as the converge point, otherwise its the diverge point. All the converge points construct the interval of convergence. All the diverge points construct the interval of divergence.
The power series is the series with form \[ \sum_{n=0}^{\infty} a_nx^n = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n+ \cdots \] where the constant \(a_0, a_1, a_2, \ldots, a_n, \ldots\) are called the coefficients of power series.
If the power series \(\displaystyle \sum_{n=1}^{\infty} a_nx^n\) is convergent at \(x=x_0\), then the power series converges for all \(|x|<x_0\). If the power series is divergent at \(x=x_0\), then the power series diverges for all \(|x|>x_0\)
Theorem. If the cofficients of power series \(\displaystyle \sum_{n=1}^{\infty} a_nx^n\) meet \(\displaystyle \lim_{n\rightarrow\infty} \left| \frac{a_{n+1}}{a_n} \right| = \rho\), then the converge radius \(R = \rho^{-1}\). Specially, we consider the \(0\) and \(+\infty\) are reciprocal to each other.
Proof. \(\displaystyle \lim_{n\rightarrow\infty} \frac{a_{n+1}x^{n+1}}{a_n x^n} = \lim_{n\rightarrow\infty} \left| \frac{a_{n+1}}{a_n} \right||x| = \rho|x|\), based on the Direct comparison test, we have
- If \(\rho|x| < 1\), that is, \(|x| \lt p^{-1}\), the series converges.
- If \(\rho|x| > 1\), that is, \(|x| \gt \rho^{-1}\), the series diverges.
By the definition of converge radius \(R\), it is exactly \(\rho^{-1}\).
For the power series with converge radius \(R>0\), then its sum function is continous in the interval of convergence, and can term-wise derivation and integration, that is,
\[ \displaystyle S'(x) = \sum_{n=0}^{\infty} (a_nx^n)' = \sum_{n=1}^{\infty} na_nx^{n-1} \]
\[ \int_0^x S(x)\dx = \sum_{n=0}^{\infty} a_n\int_0^x x^n\dx \]
The convergence will be kept during