# proof of Abel’s convergence theorem

Suppose that

$$\sum _{n=0}^{\mathrm{\infty}}{a}_{n}=L$$ |

is a convergent series^{}, and set

$$f(r)=\sum _{n=0}^{\mathrm{\infty}}{a}_{n}{r}^{n}.$$ |

Convergence of the first series implies that ${a}_{n}\to 0$, and hence $f(r)$ converges for $$. We will show that $f(r)\to L$ as $r\to {1}^{-}$.

Let

$${s}_{N}={a}_{0}+\mathrm{\cdots}+{a}_{N},N\in \mathbb{N},$$ |

denote the corresponding partial sums. Our proof relies on the following identity

$$f(r)=\sum _{n}{a}_{n}{r}^{n}=(1-r)\sum _{n}{s}_{n}{r}^{n}.$$ | (1) |

The above identity obviously works at the level of formal power series. Indeed,

$$\begin{array}{ccccccc}& \hfill {a}_{0}& \hfill +\hfill & \hfill ({a}_{1}+{a}_{0})r& \hfill +\hfill & \hfill ({a}_{2}+{a}_{1}+{a}_{0}){r}^{2}& \hfill +\mathrm{\cdots}\hfill \\ \hfill -(\hfill & & & \hfill {a}_{0}r& \hfill +\hfill & \hfill ({a}_{1}+{a}_{0}){r}^{2}& \hfill +\mathrm{\cdots})\hfill \\ \hfill =\hfill & \hfill {a}_{0}& \hfill +\hfill & \hfill {a}_{1}r& \hfill +\hfill & \hfill {a}_{2}{r}^{2}& \hfill +\mathrm{\cdots}\hfill \end{array}$$ |

Since the partial sums ${s}_{n}$ converge to $L$, they are bounded, and hence ${\sum}_{n}{s}_{n}{r}^{n}$ converges for $$. Hence for $$, identity (1) is also a genuine functional equality.

Let $\u03f5>0$ be given. Choose an $N$ sufficiently large so that all partial sums, ${s}_{n}$ with $n>N$, satisfy $|{s}_{n}-L|\le \u03f5$. Then, for all $r$ such that $$, one obtains

$$\left|\sum _{n=N+1}^{\mathrm{\infty}}({s}_{n}-L){r}^{n}\right|\le \u03f5\frac{{r}^{N+1}}{1-r}.$$ |

Note that

$$f(r)-L=(1-r)\sum _{n=0}^{N}({s}_{n}-L){r}^{n}+(1-r)\sum _{n=N+1}^{\mathrm{\infty}}({s}_{n}-L){r}^{n}.$$ |

As $r\to {1}^{-}$, the first term tends to $0$. The absolute value^{} of the
second term is estimated by $\u03f5{r}^{N+1}\le \u03f5$. Hence,

$$\underset{r\to {1}^{-}}{lim\; sup}|f(r)-L|\le \u03f5.$$ |

Since $\u03f5>0$ was arbitrary, it follows that $f(r)\to L$ as $r\to {1}^{-}$. QED

Title | proof of Abel’s convergence theorem |
---|---|

Canonical name | ProofOfAbelsConvergenceTheorem |

Date of creation | 2013-03-22 13:07:39 |

Last modified on | 2013-03-22 13:07:39 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 9 |

Author | rmilson (146) |

Entry type | Proof |

Classification | msc 40G10 |

Related topic | ProofOfAbelsLimitTheorem |