Theorem:6481641

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theorem

Named after: Niels Henrik Abel

This page was built for theorem: Abel's theorem

Suppose that $\Sigma a_n x^n$ has a radius of convergence $r$ and that $\Sigma a_n r^n$ is convergent. Then \begin{equation*} \lim_{r^-} \Sigma a_n x^n = \Sigma a_n r^n. \end{equation*}

Proof. Suppose that \[ \sum_{n=0}^{\infty} a_n = L \] is a convergent series, and define \[ f(r) = \sum_{n=0}^{\infty} a_n r^n. \]

The convergence of the first series implies that \( a_n \to 0 \), and hence \( f(r) \) converges for \( |r| < 1 \). We will show that \( f(r) \to L \) as \( r \to 1^- \).

Let \[ s_N = a_0 + a_1 + \cdots + a_N, \quad N \in \mathbb{N}, \] denote the corresponding partial sums. Our proof relies on the following identity: \[ f(r) = \sum_{n=0}^{\infty} a_n r^n = (1 - r) \sum_{n=0}^{\infty} s_n r^n. \tag{1} \] The above identity obviously works at the level of formal power series. Indeed, we can write: \[ a_0 + (a_1 + a_0)r + (a_2 + a_1 + a_0)r^2 + \cdots - \left( a_0 r + (a_1 + a_0) r^2 + \cdots \right) = a_0 + a_1 r + a_2 r^2 + \cdots. \]

Since the partial sums \( s_n \) converge to \( L \), they are bounded, and hence \[ \sum_{n=0}^{\infty} s_n r^n \] converges for \( |r| < 1 \). Hence, for \( |r| < 1 \), identity (1) is also a genuine functional equality.

Let \( \epsilon > 0 \) be given. Choose an \( N \) sufficiently large so that all partial sums \( s_n \) with \( n > N \) satisfy \[ |s_n - L| \leq \epsilon. \] Then, for all \( r \) such that \( 0 < r < 1 \), one obtains \[ \left| \sum_{n=N+1}^{\infty} (s_n - L) r^n \right| \leq \frac{\epsilon r^{N+1}}{1 - r}. \]

Note that \[ f(r) - L = (1 - r) \sum_{n=0}^{N} (s_n - L) r^n + (1 - r) \sum_{n=N+1}^{\infty} (s_n - L) r^n. \] As \( r \to 1^- \), the first term tends to 0. The absolute value of the second term is estimated by \[ \epsilon r^{N+1} \leq \epsilon. \] Hence, \[ \limsup_{r \to 1^-} |f(r) - L| \leq \epsilon. \] Since \( \epsilon > 0 \) was arbitrary, it follows that \( f(r) \to L \) as \( r \to 1^- \).