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Taylor-series representation by vanishing remainders

Statement

Let $I$ be an open interval, let $f \in C^{\infty} (I)$ , and let $a \in I$ . For each $x \in I$ , $f (x)$ equals the sum of the Taylor series of $f$ at $a$ if and only if

$R_{n, a} f (x) ⟶ 0.$

Moreover, let $r > 0$ and suppose that $[a - r, a + r] \subset I$ . For every $n \geq 0$ , set

$M_{n + 1} : = \max_{∣ t - a ∣ \leq r} ∣ f^{(n + 1)} (t) ∣ .$

$\frac{M_{n + 1} r^{n + 1}}{(n + 1)!} ⟶ 0,$

then the Taylor polynomials $T_{n, a} f$ converge uniformly to $f$ on the compact interval $[a - r, a + r]$ .

Facts & Assumptions

Given: An open interval $I$ , a function $f \in C^{\infty} (I)$ , a point $a \in I$ , and the Taylor polynomials and remainders of $f$ at $a$ .

[A1]

For the uniform assertion, $r > 0$ , $[a - r, a + r] \subset I$ , and $M_{n + 1} r^{n + 1} / (n + 1)! \to 0$ , where $M_{n + 1} = \max_{∣ t - a ∣ \leq r} ∣ f^{(n + 1)} (t) ∣$ .

[L1]

The $n$ th partial sum of the Taylor series at $a$ is $T_{n, a} f$ , and $R_{n, a} f (x) = f (x) - T_{n, a} f (x)$ (Taylor and Maclaurin series, Taylor polynomials and their remainders).

[L2]

A continuous real-valued function on a nonempty compact set attains its maximum and minimum (Extreme value theorem: a continuous real function on a nonempty compact subset of $R$ attains a greatest and a least value).

[L3]

If $∣ f^{(n + 1)} (t) ∣ \leq M$ throughout the closed interval between $a$ and $x$ , then $∣ R_{n, a} f (x) ∣ \leq \frac{M ∣ x - a ∣^{n + 1}}{(n + 1)!}$ (A uniform derivative bound gives a uniform Taylor remainder bound).

[L4]

A sequence $(g_{n})$ converges uniformly to $g$ on a set $E$ exactly when, for every $ε > 0$ , there is $N$ such that $∣ g_{n} (x) - g (x) ∣ < ε$ for every $x \in E$ and every $n \geq N$ (Pointwise convergence, uniform convergence, and the uniformly Cauchy condition for sequences of real-valued functions).

[L5]

A function is of class $C^{k}$ on an interval when $f^{(j)}$ exists there for every $j \leq k$ and each such $f^{(j)}$ is continuous there, and it is smooth, or $C^{\infty}$ , when it is $C^{k}$ for every $k \in N$ (Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ ).

[L6]

If $g$ is continuous at a point of its domain, then so is $∣ g ∣$ , the function $x \mapsto ∣ g (x) ∣$ (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function).

[L7]

For reals $u \leq v$ , the closed bounded interval $[u, v]$ is compact (Heine-Borel by bisection: every closed bounded interval $[a, b]$ is compact).

Proof

technique · direct

1.1

Fix $x \in I$ . If $T_{n, a} f (x) \to f (x)$ , then $R_{n, a} f (x) = f (x) - T_{n, a} f (x) \to 0$ .

L1assume-hypalgebra

1.2

Conversely, if $R_{n, a} f (x) \to 0$ , then $T_{n, a} f (x) = f (x) - R_{n, a} f (x) \to f (x)$ .

L1assume-hypalgebra

1.3

Let $n \geq 0$ . Since $f \in C^{\infty} (I)$ , the derivative $f^{(n + 1)}$ exists and is continuous on $I$ , so $∣ f^{(n + 1)} ∣$ is continuous on $I$ ; and $[a - r, a + r]$ is a closed bounded interval, hence compact, and it is a nonempty subset of $I$ because $r > 0$ and $a$ belongs to it. Applying the extreme value theorem to $∣ f^{(n + 1)} ∣$ on $[a - r, a + r]$ therefore produces the displayed maximum $M_{n + 1}$ .

A1L2L5L6L7

1.4

For every $y \in [a - r, a + r]$ and every $n \geq 0$ , $∣ f (y) - T_{n, a} f (y) ∣ = ∣ R_{n, a} f (y) ∣ \leq M_{n + 1} ∣ y - a ∣^{n + 1} / (n + 1)! \leq M_{n + 1} r^{n + 1} / (n + 1)!$ .

A1L1L3

1.5

Given $ε > 0$ , choose $N$ such that $M_{n + 1} r^{n + 1} / (n + 1)! < ε$ whenever $n \geq N$ .

A1choose

2.1

Thus, for this arbitrary $x$ , the Taylor series sums to $f (x)$ if and only if $R_{n, a} f (x) \to 0$ .

step 1.1step 1.2

2.2

For every $n \geq N$ and every $y \in [a - r, a + r]$ , $∣ f (y) - T_{n, a} f (y) ∣ < ε$ .

step 1.4step 1.5

3.1

Hence $T_{n, a} f \to f$ uniformly on $[a - r, a + r]$ .

L4step 2.2∎

Depends on

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Sources

W. F. Trench, Introduction to Real Analysis, §4.5, pp. 265–266 (standard reference, not scraped)
J. K. Hunter, An Introduction to Real Analysis, §10.7.1 (standard reference, not scraped)