The Cosine Series

The Cosine Series as a Function

The previous tutorial introduced the companion series that arises when the sine series is differentiated. That series has even powers, starts with the constant term $1$, and defines a function on the whole real line. We now study what its coefficients reveal about its symmetry and size. The derivative relations already established for the sine and cosine series will also lead to an identity that holds for every real input.

1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots =\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}.

Definition: For $x\in\mathbb{R}$, define $$ C(x)=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}. $$ We write $\cos x=C(x)$. This power series is called the cosine series.

The Sine Series tutorial established that this companion power series has infinite radius of convergence, so the definition makes sense for every real $x$. It also established the derivative relations

S'(x)=C(x),\qquad C'(x)=-S(x),

where $S$ denotes the sine series. We will use these relations rather than derive them again. The coefficients give $C(0)=1$ and $S(0)=0$. Differentiating the relation $C'=-S$ once more gives $C''=-S'=-C$, so the cosine series satisfies the same second-order differential equation as the sine series.

Evenness from the Powers

Every exponent in the cosine series is even. Thus changing the sign of the input leaves each term unchanged. Since the series converges for every real input, this term-by-term observation proves a symmetry of the sum function.

Proposition (Evenness of the Cosine Series): For every real $x$, $C(-x)=C(x)$.

Proof. For every integer $n\geq0$, $(-x)^{2n}=x^{2n}$. Therefore the $n$th term of the cosine series at $-x$ equals the $n$th term at $x$. Both series converge, and their corresponding partial sums are equal, so their limits are equal. Hence $C(-x)=C(x)$. $\square$

Evenness lets us transfer any estimate proved for nonnegative inputs to negative inputs. It is a feature of this particular series, not a conclusion that follows just from convergence: it comes from the parity of the exponents. The same reasoning explains why the odd powers in the sine series give an odd function.

A Conserved Identity

The derivative relations couple $S$ and $C$: the derivative of one is the other up to a sign. This coupling makes a certain combination of their squares constant. The argument is a useful general technique: differentiate a quantity built from related functions, and look for cancellation.

Theorem (Sine–Cosine Identity): For every real $x$, $$ C(x)^2+S(x)^2=1. $$ Consequently, $|C(x)|\leq1$ and $|S(x)|\leq1$ for every real $x$.

Proof. Define $F(x)=C(x)^2+S(x)^2$. Since $S$ and $C$ are differentiable, the chain rule and the derivative relations give

$$ F'(x)=2C(x)C'(x)+2S(x)S'(x) =2C(x)(-S(x))+2S(x)C(x)=0. $$

For any two real numbers $u<v$, the Mean Value Theorem applied to $F$ on $[u,v]$ gives a point $w\in(u,v)$ such that $F(v)-F(u)=F'(w)(v-u)$. Since $F'(w)=0$, it follows that $F(v)=F(u)$. Thus $F$ is constant on the real line. Evaluating at zero,

$$ F(0)=C(0)^2+S(0)^2=1^2+0^2=1. $$

Therefore $F(x)=1$ for every real $x$, which proves the identity. Since both squares are nonnegative, $C(x)^2\leq1$ and $S(x)^2\leq1$; taking square roots yields the stated bounds. $\square$

The identity is a global conclusion derived from the series and their derivatives. In particular, it does not assume a geometric interpretation of sine or cosine. The derivative calculation proves that the sum of the squares cannot change, and the initial values determine its constant value.

Alternating-Series Estimates Near Zero

The global identity bounds the absolute values of $S$ and $C$, but it does not by itself say whether $C(x)$ is positive near zero. For that, the cosine series gives a sharper estimate. When $|x|\leq1$, the absolute terms

u_n=\frac{|x|^{2n}}{(2n)!}

are nonincreasing. For $x\neq0$, their consecutive ratio is

\frac{u_{n+1}}{u_n} =\frac{|x|^2}{(2n+1)(2n+2)} \leq\frac{1}{2} \qquad(n\geq0),

and when $x=0$ the series is simply $C(0)=1$. The Alternating Series Test and its remainder bounds therefore apply. The partial sum through index $n=1$ is a lower bound, and the partial sum through index $n=2$ is an upper bound. Thus, for $|x|\leq1$,

Corollary (Cosine Estimate Near Zero): For every real $x$ with $|x|\leq1$, $$ 1-\frac{x^2}{2}\leq C(x)\leq1-\frac{x^2}{2}+\frac{x^4}{24}\leq1. $$ In particular, $C(x)\geq\frac12>0$ on this interval.

Proof. The alternating-series bounds give the first two inequalities. For the last one, $x^4\leq x^2$ when $|x|\leq1$, so

1-\frac{x^2}{2}+\frac{x^4}{24} \leq1-\frac{x^2}{2}+\frac{x^2}{24} =1-\frac{11x^2}{24}\leq1.

Also $1-x^2/2\geq1-1/2=1/2$, proving the positivity assertion. $\square$

The final bound $C(x)\leq1$ also follows from the Sine–Cosine Identity, but the alternating-series estimate gives more information: it locates $C(x)$ within a short interval whose width is $x^4/24$. This is useful when an explicit approximation is needed.

Worked Example: Enclosing the Value at One

At $x=1$, the lower bound from the first two terms is

1-\frac{1^2}{2}=\frac12.

The next partial sum adds the positive term $1/4!=1/24$, giving

1-\frac{1}{2}+\frac{1}{24} =\frac{12}{24}+\frac{1}{24} =\frac{13}{24}.

The alternating-series bounds therefore yield

\frac12\leq C(1)\leq\frac{13}{24}.

In particular, the series proves $C(1)>0$, and it places its value in an interval of width $13/24-1/2=1/24$. Evenness gives the same enclosure for $C(-1)$.

Worked Example: Estimating the Value at Negative One-Half

By evenness, $C(-1/2)=C(1/2)$. The near-zero estimate applies because $|1/2|\leq1$. Its lower bound is

1-\frac{(1/2)^2}{2} =1-\frac18 =\frac78.

Its upper bound adds $(1/2)^4/24=1/384$, so

C\left(-\frac12\right) =C\left(\frac12\right) \leq\frac78+\frac{1}{384} =\frac{336}{384}+\frac{1}{384} =\frac{337}{384}.

Consequently,

\frac78\leq C\left(-\frac12\right)\leq\frac{337}{384}.

The lower bound is positive, and the width of this enclosure is $337/384-7/8=1/384$. The estimate uses both the evenness of the series and the alternating-series remainder bound.

Worked Example: A Geometric Tail Bound at Two

For larger inputs, the absolute terms need not decrease from the beginning, so the alternating-series remainder bound cannot be applied without checking its hypothesis. At $x=2$, the partial sum through $n=3$ is

1-\frac{2^2}{2!}+\frac{2^4}{4!}-\frac{2^6}{6!} =1-2+\frac23-\frac4{45} =-\frac{19}{45}.

For $n\geq4$, set $u_n=2^{2n}/(2n)!$. These terms satisfy

\frac{u_{n+1}}{u_n} =\frac{4}{(2n+1)(2n+2)} \leq\frac{4}{9\cdot10} =\frac{2}{45}.

The first omitted absolute term is $u_4=2^8/8!=2/315$. Repeated application of the ratio bound gives $u_{4+j}\leq(2/315)(2/45)^j$ for every integer $j\geq0$. Hence the absolute value of the tail is at most

\sum_{n=4}^{\infty}u_n \leq\frac{2}{315}\sum_{j=0}^{\infty}\left(\frac{2}{45}\right)^j =\frac{2}{315}\cdot\frac{45}{43} =\frac{2}{301}.

Therefore the value of the full series differs from the displayed partial sum by at most $2/301$:

-\frac{19}{45}-\frac{2}{301} \leq C(2)\leq -\frac{19}{45}+\frac{2}{301}.

This estimate does not rely on the terms decreasing from $n=0$. It uses only the verified ratio bound on the tail starting at $n=4$.

What the Cosine Series Reveals

The cosine series is even because all its powers are even. Together with the sine series, it satisfies the identity $C(x)^2+S(x)^2=1$, proved by differentiating the sum of the squares and using the initial values at zero. This identity gives global bounds, while the alternating-series estimates give a more precise description near zero.

A common pitfall is to use the alternating-series remainder bound merely because a series alternates. The bound also requires that the absolute terms be nonincreasing and tend to zero. For $|x|\leq1$, the ratio calculation verifies the required decrease at every index. For a larger input such as $x=2$, it is safer to find an index beyond which the ratios are uniformly less than one and then bound the tail geometrically.

The distinction between a global fact and a local estimate is also important. The Sine–Cosine Identity gives $|C(x)|\leq1$ for every real $x$, but it does not alone establish positivity. The alternating-series estimate proves positivity on $[-1,1]$ by showing that $C(x)\geq1/2$ there. Each conclusion depends on its own argument.

Takeaway: The cosine series defines an even function, and the derivative relations with the sine series imply $C(x)^2+S(x)^2=1$. Thus both functions are globally bounded in absolute value by one, while alternating-series estimates show that $C(x)$ is positive and close to $1-x^2/2$ when $|x|\leq1$.

Check Your Understanding

Use the cosine series, its derivative relation with the sine series, and the estimates proved above to answer the following questions.

Why does the even-power structure of the cosine series imply $C(-x)=C(x)$?
Which derivative relations make the derivative of $C(x)^2+S(x)^2$ equal to zero?
How does evaluating $C(x)^2+S(x)^2$ at zero determine its value for every real $x$?
What condition on the absolute terms is needed before applying the alternating-series remainder bound?
What lower bound does the cosine series give when $|x|\leq1$, and what does it imply about the sign of $C(x)$ on that interval?

What You'll Learn