Tauber's theorem

Tauber 's theorem is a theorem on the properties of power series near the boundary of a circle of convergence . It is the simplest inverse theorem to the Abel theorem on the convergence of power series. It is proved by in 1897. ^[1] Subsequently, it was formulated and proved under more general conditions by other authors ( Abel – Tauber Theorem ).

Content

1 Formulation
2 Explanations
3 Proof
4 Lemma
5 notes
6 Literature

Wording

If $a_{n}=o({\frac {1}{n}})$ ${\ displaystyle a_ {n} = o ({\ frac {1} {n}})}$ and $\sum _{n=0}^{\infty }a_{n}x^{n}=s$ ${\ displaystyle \ sum _ {n = 0} ^ {\ infty} a_ {n} x ^ {n} = s}$ at $x\to 1$ ${\ displaystyle x \ to 1}$ left then row $\sum _{n=0}^{\infty }a_{n}$ ${\ displaystyle \ sum _ {n = 0} ^ {\ infty} a_ {n}}$ converges to the sum $s$ ${\ displaystyle s}$ .

Explanation

Here equality $f(x)=o(\varphi (x))$ ${\ displaystyle f (x) = o (\ varphi (x))}$ means that ${\frac {f(x)}{\varphi (x)}}\rightarrow 0$ ${\ displaystyle {\ frac {f (x)} {\ varphi (x)}} \ rightarrow 0}$ when $x$ ${\ displaystyle x}$ tends to a given limit (see O-notation ).

Proof

It is enough to prove that for $N=[{\frac {1}{(1-x)}}]$ ${\ displaystyle N = [{\ frac {1} {(1-x)}}]}$ and $x\rightarrow 1$ ${\ displaystyle x \ rightarrow 1}$ performed

\sum _{n=0}^{\infty }a_{n}x^{n}-\sum _{n=0}^{N}a_{n}\rightarrow 0

{\ displaystyle \ sum _ {n = 0} ^ {\ infty} a_ {n} x ^ {n} - \ sum _ {n = 0} ^ {N} a_ {n} \ rightarrow 0}

.

i.e

\sum _{n=N+1}^{\infty }a_{n}x^{n}-\sum _{n=0}^{N}a_{n}(1-x^{n})\rightarrow 0

{\ displaystyle \ sum _ {n = N + 1} ^ {\ infty} a_ {n} x ^ {n} - \ sum _ {n = 0} ^ {N} a_ {n} (1-x ^ { n}) \ rightarrow 0}

.

Denote:

S_{1}=\sum _{n=N+1}^{\infty }a_{n}x^{n}

{\ displaystyle S_ {1} = \ sum _ {n = N + 1} ^ {\ infty} a_ {n} x ^ {n}}

,

S_{2}=\sum _{n=0}^{N}a_{n}(1-x^{n})

{\ displaystyle S_ {2} = \ sum _ {n = 0} ^ {N} a_ {n} (1-x ^ {n})}

.

Obviously:

\left|S_{1}\right|=\left|\sum _{n=N+1}^{\infty }na_{n}{\frac {x^{n}}{n}}\right|<{\frac {\epsilon }{N+1}}\sum _{n=N+1}^{\infty }x^{n}<{\frac {\epsilon }{(N+1)(1-x)}}<\epsilon

{\ displaystyle \ left | S_ {1} \ right | = \ left | \ sum _ {n = N + 1} ^ {\ infty} na_ {n} {\ frac {x ^ {n}} {n}} \ right | <{\ frac {\ epsilon} {N + 1}} \ sum _ {n = N + 1} ^ {\ infty} x ^ {n} <{\ frac {\ epsilon} {(N + 1 ) (1-x)}} <\ epsilon}

.

Due to the fact that

1-x^{n}=(1-x)(1+x+...+x^{n-1})<n(1-x)

{\ displaystyle 1-x ^ {n} = (1-x) (1 + x + ... + x ^ {n-1}) <n (1-x)}

follows:

\left|S_{2}\right|<(1-x)\sum _{n=0}^{N}n\left|a_{n}\right|\leqslant {\frac {1}{N}}\sum _{n=0}^{N}n\left|a_{n}\right|

{\ displaystyle \ left | S_ {2} \ right | <(1-x) \ sum _ {n = 0} ^ {N} n \ left | a_ {n} \ right | \ leqslant {\ frac {1} {N}} \ sum _ {n = 0} ^ {N} n \ left | a_ {n} \ right |}

.

By the lemma, the right-hand side tends to zero, so that and $\left|S_{2}\right|<\epsilon$ ${\ displaystyle \ left | S_ {2} \ right | <\ epsilon}$ at sufficiently large $N$ ${\ displaystyle N}$ we get $\left|S_{1}-S_{2}\right|<2\epsilon$ ${\ displaystyle \ left | S_ {1} -S_ {2} \ right | <2 \ epsilon}$ . The proof of the theorem is complete.

Lemma

If $b_{n}\rightarrow 0$ ${\ displaystyle b_ {n} \ rightarrow 0}$ at $n\rightarrow \infty$ ${\ displaystyle n \ rightarrow \ infty}$ then ${\frac {b_{0}+b_{1}+...+b_{n}}{n+1}}\rightarrow 0$ ${\ displaystyle {\ frac {b_ {0} + b_ {1} + ... + b_ {n}} {n + 1}} \ rightarrow 0}$ .

You can always find such numbers $K$ ${\ displaystyle K}$ , $\epsilon$ ${\ displaystyle \ epsilon}$ , $n_{0}$ ${\ displaystyle n_ {0}}$ , what $\left|b_{n}\right|<K$ ${\ displaystyle \ left | b_ {n} \ right | <K}$ for all $n$ ${\ displaystyle n}$ and $\left|b_{n}\right|<\epsilon$ ${\ displaystyle \ left | b_ {n} \ right | <\ epsilon}$ at $n>n_{0}$ ${\ displaystyle n> n_ {0}}$ .

Take $n>n_{0}$ ${\ displaystyle n> n_ {0}}$ and $n>(n_{0}+1){\frac {K}{\epsilon }}$ ${\ displaystyle n> (n_ {0} +1) {\ frac {K} {\ epsilon}}}$ .

We have:

\left|{\frac {b_{0}+b_{1}+...+b_{n}}{n+1}}\right|\leqslant \left|{\frac {b_{0}+b_{1}+...+b_{n_{0}}}{n+1}}\right|+\left|{\frac {b_{n_{0}+1}+b_{1}+...+b_{n}}{n+1}}\right|\leqslant {\frac {(n_{0}+1)K}{n+1}}+{\frac {(n-n_{0})\epsilon }{n+1}}<2\epsilon

{\ displaystyle \ left | {\ frac {b_ {0} + b_ {1} + ... + b_ {n}} {n + 1}} \ right | \ leqslant \ left | {\ frac {b_ {0 } + b_ {1} + ... + b_ {n_ {0}}} {n + 1}} \ right | + \ left | {\ frac {b_ {n_ {0} +1} + b_ {1} + ... + b_ {n}} {n + 1}} \ right | \ leqslant {\ frac {(n_ {0} +1) K} {n + 1}} + {\ frac {(n-n_ {0}) \ epsilon} {n + 1}} <2 \ epsilon}

.

The proof of the lemma is complete.

Notes

↑ Tauber, A. Ein Satz aus der Theorie der unendlichen Reihen (A theorem from the theory of infinite series) // Monatsh. F. Math. - 1897. - V. 8. - S. 273-277. - DOI 10.1007 / BF01696278

Literature

E. Titchmarsh. Function theory. - Science, 1980 .-- 464 p.

[1] Tauber, A. Ein Satz aus der Theorie der unendlichen Reihen (A theorem from the theory of infinite series) // Monatsh. F. Math. - 1897. - V. 8. - S. 273-277. - DOI 10.1007 / BF01696278