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Standard Maclaurin expansions

Statement

The standard Maclaurin expansions are

$\frac{1}{1 - x} = \sum_{n = 0}^{\infty} x^{n} (∣ x ∣ < 1),$

$e^{x} = \sum_{n = 0}^{\infty} \frac{x^{n}}{n!}, \sin x = \sum_{n = 0}^{\infty} \frac{(- 1)^{n} x^{2 n + 1}}{(2 n + 1)!}, \cos x = \sum_{n = 0}^{\infty} \frac{(- 1)^{n} x^{2 n}}{(2 n)!} (x \in R),$

$\log (1 + x) = \sum_{n = 1}^{\infty} \frac{(- 1)^{n + 1} x^{n}}{n} (- 1 < x \leq 1),$

and

$\arctan x = \sum_{n = 0}^{\infty} \frac{(- 1)^{n} x^{2 n + 1}}{2 n + 1} (∣ x ∣ < 1),$

with value $π / 4$ at $x = 1$ . For every real $α$ ,

$(1 + x)^{α} = \sum_{n = 0}^{\infty} (\binom{α}{n}) x^{n} (∣ x ∣ < 1),$

where

$(\binom{α}{0}) = 1, (\binom{α}{n + 1}) = (\binom{α}{n}) \frac{α - n}{n + 1} .$

Facts & Assumptions

Given: The functions and power series displayed in the statement, and an arbitrary real parameter $α$ for the generalized-binomial family.

[L1]

The Maclaurin series of a smooth function $f$ is $\sum_{n \geq 0} f^{(n)} (0) x^{n} / n!$ (Taylor and Maclaurin series).

[L2]

For $∣ x ∣ < 1$ , the geometric series satisfies $\sum_{n \geq 0} x^{n} = 1 / (1 - x)$ (For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges).

[L3]

For every real $x$ , $\exp (x) = \sum_{n \geq 0} x^{n} / n!$ , and $e : = \exp (1)$ (The real exponential function and the number $e$ by a power series); and $\sin x = \sum_{n \geq 0} (- 1)^{n} x^{2 n + 1} / (2 n + 1)!$ , $\cos x = \sum_{n \geq 0} (- 1)^{n} x^{2 n} / (2 n)!$ (Sine and cosine defined by their real power series). The first definition names $\exp$ and names $e$ only as $\exp (1)$ ; it does not name $e^{x}$ .

[L4]

For $- 1 < x \leq 1$ , $\log (1 + x) = \sum_{n \geq 1} (- 1)^{n + 1} x^{n} / n$ ; at $x = 1$ the sum is $\log 2$ , and at $x = - 1$ the series diverges ([The power series for log(1+x) on (-1,1], including the Abel endpoint](/item/thm-log-one-plus-x-power-series)).

[L5]

For $∣ x ∣ < 1$ , $\arctan x = \sum_{n \geq 0} (- 1)^{n} x^{2 n + 1} / (2 n + 1)$ , and the series at $x = 1$ converges to $π / 4$ (Principal arctangent: derivative, integral, power series, and the Gregory–Leibniz series).

[L6]

For every real $β$ , $x \mapsto x^{β}$ is continuous on $(0, \infty)$ and has derivative $β x^{β - 1}$ there (Continuity and derivatives of positive-base real powers).

[L7]

Factorials satisfy $(n + 1)! = (n + 1) n!$ (The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ ), and for reals $u, v$ and a natural number $m$ the finite binomial theorem is $(u + v)^{m} = \sum_{k = 0}^{m} (\binom{m}{k}) u^{k} v^{m - k}$ (The binomial theorem in $R$ : $(x + y)^{n} = \sum_{k < n + 1} ι ⁣ (\binom{n}{k}) x^{k} y^{n - k}$ ), the coefficients being the natural numbers $(\binom{m}{k})$ read in $R$ through the canonical embedding. The exponents there are the natural-number ones: for every real $a$ , $n \mapsto a^{n}$ is the unique function on $N$ with $a^{0} = 1$ and $a^{n + 1} = a^{n} \cdot a$ (Integer powers $a^{m}$ ).

[L8]

If a sequence has no zero terms and ${lim sup}_{n \to \infty} ∣ a_{n + 1} / a_{n} ∣ < 1$ , then $\sum a_{n}$ converges absolutely (Ratio test: $lim sup ∣ a_{k + 1} / a_{k} ∣ < 1$ gives absolute convergence and hence convergence, and $lim inf ∣ a_{k + 1} / a_{k} ∣ > 1$ gives divergence).

[L9]

A real power series may be differentiated term by term at every point inside its radius of convergence, and the differentiated series has the same radius (Inside its radius a real power series may be differentiated term by term, and the differentiated series has the same radius).

[L10]

Sums and products of differentiable functions satisfy the sum and product rules (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[L11]

If a real-valued function is continuous on an order-convex interval and has derivative zero at every interior point, then it is constant (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant).

[L12]

For naturals $k \leq n$ , $(\binom{n}{k}) k! (n - k)! = n!$ ; consequently, as a real number, $(\binom{n}{k}) = n! / (k! (n - k)!)$ ( $(\binom{n}{k}) k! (n - k)! = n!$ for $k \leq n$ ; hence $(\binom{n}{k}) k! = n^{\underline{k}}$ , the quotient $n! / (k! (n - k)!)$ is a natural number, and $(\binom{n}{k}) = (\binom{n}{n - k})$ ).

[L13]

If $g$ is differentiable at a limit point $c$ of its domain and $h$ is differentiable at $g (c)$ , itself a limit point of the domain of $h$ , then $h \circ g$ is differentiable at $c$ and $(h \circ g)^{'} (c) = h^{'} (g (c)) g^{'} (c)$ (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ).

[L14]

For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable at every real with derivative $n x^{n - 1}$ , and for $n = 0$ it is the constant $1$ , with derivative $0$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term (For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term).

[L15]

A function differentiable at a limit point $c$ of its domain is continuous at $c$ (A function differentiable at $c$ is continuous at $c$ ).

[L16]

For $a > 0$ and $x \in R$ , $a^{x} = \exp (x \log a)$ (Real powers for positive bases, with the zero-base positive-exponent convention); $\log$ is the inverse of $\exp$ , so $\log (\exp y) = y$ for every real $y$ and $\exp (\log w) = w$ for every $w > 0$ (The natural logarithm as the inverse of the exponential function); and for $a, b > 0$ and $r, s \in R$ , $a^{r + s} = a^{r} a^{s}$ (The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents).

[L17]

$\exp : R \to (0, \infty)$ is a bijection (The exponential is a continuous bijection from $R$ onto $(0, \infty)$ ); in particular every value of $\exp$ is positive.

[L18]

A real power series about the centre $c$ is $\sum_{n \geq 0} a_{n} (x - c)^{n}$ , where the powers are those of Integer powers $a^{m}$ and convergence is that of Series, partial sums, convergence and the sum, divergence, and the tail series; at $x = c$ it always converges to $a_{0}$ , the $n = 0$ term being $a_{0}$ because $0^{0} = 1$ and every later term being $0$ ; and its radius of convergence is the supremum, in $[0, + \infty]$ , of the $r \geq 0$ such that the series converges absolutely at every $x$ with $∣ x - c ∣ < r$ (A real power series about a centre, its interval of convergence, and its radius in $[0, + \infty]$ ).

[L19]

$f^{(0)} : = f$ , and $f^{(j + 1)} : = (f^{(j)})^{'}$ wherever $f^{(j)}$ is differentiable; $f$ is smooth, that is $C^{\infty}$ , on an interval when every $f^{(j)}$ exists there and is continuous there (Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ ).

[L20]

For naturals $n, k$ , $(\binom{n}{k})$ is the number of $k$ -element subsets of $n$ ; in particular $(\binom{n}{0}) = 1$ for every $n$ , and $(\binom{n}{k}) = 0$ for $k > n$ (The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ ).

[L21]

If $\sum a_{k}$ and $\sum b_{k}$ converge and $c$ is real, then $\sum (a_{k} + b_{k})$ converges to $\sum a_{k} + \sum b_{k}$ , and $\sum c a_{k}$ converges to $c \sum a_{k}$ (Convergent series add and scale termwise).

[L22]

A real sequence converges to a real $L$ if and only if its limit inferior and its limit superior are both $L$ (A real sequence converges to $L \in R$ iff $lim inf x_{k} = lim sup x_{k} = L$ , and diverges to $\pm \infty$ iff both equal $\pm \infty$ ).

[L23]

If $x_{k} \to x$ and $y_{k} \to y$ then $x_{k} + y_{k} \to x + y$ , $c x_{k} \to c x$ , $x_{k} - y_{k} \to x - y$ and $x_{k} y_{k} \to x y$ (Algebra of limits: sums, scalar multiples, products and quotients); and for every $ε > 0$ there is a natural $n \geq 1$ with $1 / n < ε$ (For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ ).

[L24]

A real power series $\sum a_{n} (x - c)^{n}$ of radius $R$ converges absolutely at every $x$ with $∣ x - c ∣ < R$ and diverges at every $x$ with $∣ x - c ∣ > R$ (A real power series converges absolutely inside its radius and diverges outside it, while either behaviour may occur at an endpoint).

Proof

technique · direct

1.1

The geometric, sine, cosine, logarithmic, and inverse-tangent identities, with exactly the displayed domains and endpoint assertions, are [L2]–[L5]. The exponential identity needs one further move, because the statement writes $e^{x}$ , the real power of the base $e$ in the sense of [L16], while [L3] defines $\exp$ and defines $e$ only as $\exp (1)$ . By [L17] every value of $\exp$ is positive, so $e = \exp (1) > 0$ and [L16] assigns $e^{x}$ the value $\exp (x \log e)$ . Since $\log$ is the inverse of $\exp$ [L16], $\log e = \log (\exp 1) = 1$ , so $e^{x} = \exp (x \cdot 1) = \exp (x) (x \in R),$ and the series [L3] gives for $\exp (x)$ is therefore the series of $e^{x}$ .

L2L3L4L5L16L17algebra

1.2

Suppose $f (x) = \sum_{n \geq 0} a_{n} x^{n}$ for $∣ x ∣ < R$ with $R > 0$ ; this is a power series about $0$ in the sense of [L18], and the hypothesis is exactly that it converges, with sum $f (x)$ , at every $x$ with $∣ x ∣ < R$ . Its radius of convergence $ρ$ is therefore at least $R$ : by [L24] a power series diverges at every point farther from its centre than its radius, so a point of convergence $x$ has $∣ x ∣ \leq ρ$ , and letting $∣ x ∣$ run over $[0, R)$ gives $ρ \geq R$ . Every point of $(- R, R)$ thus lies strictly inside the radius, so by [L9] the sum is differentiable there and its derivative is again a power series of radius $ρ$ ; induction on $k$ , using $(n + 1)! = (n + 1) n!$ from [L7] at each step, gives $f^{(k)} (x) = \sum_{n \geq 0} \frac{(n + k)!}{n!} a_{n + k} x^{n} (∣ x ∣ < R)$ for every $k \geq 0$ , each again a power series about $0$ of radius $ρ$ . Evaluating at the centre, [L18] gives $f^{(k)} (0) = k! a_{k}$ . Every $f^{(k)}$ is differentiable on $(- R, R)$ , hence continuous there by [L15], so $f$ is smooth on $(- R, R)$ in the sense of [L19] and [L1] applies at $a = 0$ : the Maclaurin series of $f$ is $\sum_{k \geq 0} f^{(k)} (0) x^{k} / k! = \sum_{k \geq 0} a_{k} x^{k}$ . A power series representing $f$ near $0$ is therefore its Maclaurin series.

L1L7L9L15L18L19L24algebra

1.3

Fix $α \in R$ and define $c_{0} = 1$ and $c_{n + 1} = c_{n} (α - n) / (n + 1)$ for $n \geq 0$ .

L7choose

1.4

On $(- 1, 1)$ , the function $x \mapsto (1 + x)^{- α}$ is differentiable and has derivative $- α (1 + x)^{- α - 1}$ . The inner map $x \mapsto 1 + x$ is a polynomial, so by [L14] it is differentiable with derivative $1$ at every point of $(- 1, 1)$ , each of which is a limit point of $(- 1, 1)$ ; its value $1 + x$ lies in $(0, \infty)$ and is a limit point of $(0, \infty)$ , where by [L6] the outer map $u \mapsto u^{- α}$ is differentiable with derivative $- α u^{- α - 1}$ . The chain rule [L13] therefore gives the composite derivative $- α (1 + x)^{- α - 1} \cdot 1$ .

L6L13L14algebra

2.1

If $α = m$ is a nonnegative integer, the recurrence of step 1.3 gives $c_{n} = (\binom{m}{n})$ for $0 \leq n \leq m$ and $c_{n} = 0$ for $n > m$ . Indeed $c_{0} = 1 = (\binom{m}{0})$ by [L20]; and if $n < m$ and $c_{n} = (\binom{m}{n}) = m! / (n! (m - n)!)$ as a real number by [L12], then, since $(n + 1)! = (n + 1) n!$ and $(m - n)! = (m - n) (m - n - 1)!$ with $m - n \geq 1$ , $c_{n + 1} = \frac{m!}{n! (m - n)!} \cdot \frac{m - n}{n + 1} = \frac{m!}{(n + 1)! (m - (n + 1))!} = (\binom{m}{n + 1});$ while $c_{m + 1} = c_{m} (m - m) / (m + 1) = 0$ , after which the recurrence keeps every term $0$ . Taking $u = x$ and $v = 1$ in the finite binomial theorem of [L7] then gives $(1 + x)^{m} = (x + 1)^{m} = \sum_{k = 0}^{m} (\binom{m}{k}) x^{k} 1^{m - k} = \sum_{k = 0}^{m} (\binom{m}{k}) x^{k} = \sum_{n \geq 0} c_{n} x^{n},$ using $1^{j} = 1$ for every natural $j$ , which the recursion $1^{0} = 1$ , $1^{j + 1} = 1^{j} \cdot 1$ of [L7] gives by induction, and using $c_{n} = 0$ for $n > m$ . The exponent in $(1 + x)^{m}$ is there the natural-number one of [L7]. For $∣ x ∣ < 1$ that value is also the real power $(1 + x)^{m}$ of [L16]: since $1 + x > 0$ , the real powers $(1 + x)^{n}$ with $n \in N$ satisfy $(1 + x)^{0} = \exp (0 \cdot \log (1 + x)) = \exp 0 = 1$ , the value $\exp 0 = 1$ being the value at the centre of the series [L3] gives for $\exp$ , by [L18]; and, by the addition law of [L16] together with $(1 + x)^{1} = \exp (\log (1 + x)) = 1 + x$ , also $(1 + x)^{n + 1} = (1 + x)^{n} (1 + x)^{1} = (1 + x)^{n} (1 + x)$ . That is the recursion determining the natural-number powers in [L7], so the two readings of $(1 + x)^{m}$ agree.

L3L7L12L16L18L20step 1.3algebra

2.2

Suppose $α$ is not a nonnegative integer and $0 < ∣ x ∣ < 1$ . Then, by step 1.3, $c_{0} = 1$ and no factor $α - n$ of the recurrence vanishes, so every $c_{n}$ is nonzero, and $x^{n} \neq 0$ ; the ratios $q_{n} : = ∣ c_{n + 1} x^{n + 1} / (c_{n} x^{n}) ∣ = ∣ x ∣ ∣ α - n ∣ / (n + 1)$ are therefore defined. For every $n > ∣ α ∣$ one has $∣ α - n ∣ = n - α$ and hence $q_{n} = ∣ x ∣ (1 - \frac{α + 1}{n + 1}),$ so, since $1 / (n + 1) \to 0$ by [L23] and convergence of a sequence is a condition on its tails, [L23] gives $q_{n} \to ∣ x ∣$ . By [L22] the limit superior is that same limit, so ${lim sup}_{n} q_{n} = ∣ x ∣ < 1$ and [L8] makes $\sum_{n \geq 0} c_{n} x^{n}$ converge absolutely.

L8L22L23step 1.3algebra

2.3

Thus the six series in step 1.1 are precisely the asserted Maclaurin expansions; the logarithmic and inverse-tangent endpoint values are values of the same series, not claims of an open interval beyond its radius.

step 1.1step 1.2

3.1

Therefore, for every real $α$ , the power series $B (x) : = \sum_{n \geq 0} c_{n} x^{n}$ built from step 1.3 converges absolutely at every $x$ with $∣ x ∣ < 1$ : when $α = m$ is a nonnegative integer every term past index $m$ vanishes by step 2.1 and the series is a finite sum; at $x = 0$ it converges to $c_{0}$ by [L18]; and every remaining case is step 2.2. Its radius of convergence in the sense of [L18] is therefore at least $1$ .

L18step 1.3step 2.1step 2.2algebra

4.1

By [L9] and step 3.1, $B$ is differentiable on $(- 1, 1)$ with $B^{'} (x) = \sum_{n \geq 0} (n + 1) c_{n + 1} x^{n}$ , again a power series of radius at least $1$ . The recurrence of step 1.3 gives $(n + 1) c_{n + 1} = (α - n) c_{n}$ , so $B^{'} (x) = \sum_{n \geq 0} (α - n) c_{n} x^{n}$ for $∣ x ∣ < 1$ . Both $\sum_{n \geq 0} α c_{n} x^{n}$ and $\sum_{n \geq 0} n c_{n} x^{n} = x \sum_{n \geq 0} (n + 1) c_{n + 1} x^{n}$ converge there, the first by step 3.1 and the second because it is $x B^{'} (x)$ , so [L21] splits the sum termwise into $α B (x) - x B^{'} (x)$ . Hence $B^{'} (x) = α B (x) - x B^{'} (x)$ , that is $(1 + x) B^{'} (x) = α B (x)$ , for $∣ x ∣ < 1$ .

L9L21step 1.3step 3.1algebra

5.1

For $G (x) : = (1 + x)^{- α} B (x)$ , both factors are differentiable on $(- 1, 1)$ by step 1.4 and step 4.1, so the product rule gives $G^{'} (x) = - α (1 + x)^{- α - 1} B (x) + (1 + x)^{- α} B^{'} (x)$ there. On $(- 1, 1)$ the base $1 + x$ is positive, so $(1 + x)^{1} = \exp (\log (1 + x)) = 1 + x$ and the addition law at the real exponents $- α - 1$ and $1$ gives $(1 + x)^{- α - 1} (1 + x) = (1 + x)^{- α}$ , both by [L16]. Multiplying the displayed derivative by $1 + x$ and using $(1 + x) B^{'} (x) = α B (x)$ from step 4.1 therefore gives $(1 + x) G^{'} (x) = - α (1 + x)^{- α} B (x) + α (1 + x)^{- α} B (x) = 0$ ; since $1 + x \neq 0$ , $G^{'} (x) = 0$ throughout $(- 1, 1)$ .

L10L16step 1.4step 4.1algebra

6.1

Step 5.1 makes $G$ differentiable at every point of $(- 1, 1)$ , and every such point is a limit point of $(- 1, 1)$ , so [L15] makes $G$ continuous on $(- 1, 1)$ .

L15step 5.1

7.1

The interval $(- 1, 1)$ is order-convex, so [L11] and step 6.1 make $G$ constant there, and its constant value is $G (0) = 1^{- α} B (0)$ . Here $B (0) = c_{0}$ by [L18], the value of a power series at its centre being its constant coefficient, and $c_{0} = 1$ by step 1.3. And $1^{- α}$ is a real power of the base $1 > 0$ , so [L16] makes it $\exp (- α \log 1)$ , where $\exp 0 = 1$ is the value at the centre of the series [L3] gives for $\exp$ , by [L18], hence $\log 1 = \log (\exp 0) = 0$ and $1^{- α} = \exp 0 = 1$ . The constant value is therefore $1$ , that is $(1 + x)^{- α} B (x) = 1$ for every $∣ x ∣ < 1$ . Multiplying by the real power $(1 + x)^{α}$ and using the addition law of [L16] at the real exponents $α$ and $- α$ , which gives $(1 + x)^{α} (1 + x)^{- α} = (1 + x)^{0} = \exp (0 \cdot \log (1 + x)) = 1$ , yields $B (x) = (1 + x)^{α}$ for every $∣ x ∣ < 1$ .

L3L11L16L18step 1.3step 6.1algebra

8.1

By step 1.2, applied to $B$ on $(- 1, 1)$ , this is the Maclaurin expansion of $(1 + x)^{α}$ ; its coefficients are the recursively defined numbers $(\binom{α}{n}) = c_{n}$ of the statement. That notation extends the library's binomial coefficient rather than clashing with it: when $α = m$ is a nonnegative integer, step 2.1 identifies $c_{n}$ with the count $(\binom{m}{n})$ of [L20]. The argument makes no assertion at $x = 1$ or $x = - 1$ .

L1L20step 1.2step 2.1step 3.1step 7.1

9.1

Combining steps 2.3 and 8.1 proves all the displayed expansions with no additional endpoint claims.

step 2.3step 8.1∎

Remarks

Which symbol is which. Three symbols in the statement name objects the library builds separately, and each is matched to its own definition rather than to a near neighbour.

$e^{x}$ is the real power of Real powers for positive bases, with the zero-base positive-exponent convention, not the series that defines $\exp$ : The real exponential function and the number $e$ by a power series defines $\exp$ and defines $e$ only as $\exp (1)$ . Step 1.1 supplies the bridge, from $\log e = 1$ .

The exponent in $(1 + x)^{α}$ is real, so that power too is $\exp (α \log (1 + x))$ , whereas the exponent in the finite binomial theorem is a natural number and the $(1 + x)^{m}$ appearing there is the integer power of Integer powers $a^{m}$ . Step 2.1 proves the two readings agree on $(- 1, 1)$ ; without that they are different functions with the same name.

For real $α$ the symbol $(\binom{α}{n})$ is defined by the recurrence in the statement, while $(\binom{n}{k})$ elsewhere in the library is a count (The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ ). Step 2.1 proves the recurrence reproduces the count at a nonnegative integer $α$ , so the notation extends rather than overloads.

Two power conventions coexist here without conflict, and it is worth saying which is used where. The integer power fixes $0^{0} = 1$ , which is what makes a power series equal its constant coefficient at the centre; the real power leaves $0^{0}$ undefined and is only ever applied above to bases $1 + x > 0$ , $1$ , and $e$ .

Depends on

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Sources

W. F. Trench, Introduction to Real Analysis, §4.5, pp. 265–267 (standard reference, not scraped)
MIT OpenCourseWare 18.100C Real Analysis, Lecture 23 (standard reference, not scraped)