Cosine Function is Continuous

From ProofWiki
Jump to: navigation, search

Theorem

Let $x \in \R$ be a real number.

Let $\cos x$ be the cosine of $x$.


Then:

$\cos x$ is continuous on $\R$.


Proof

Recall the definition of the cosine function:

$\displaystyle \cos x = \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac {x^{2n}}{\left({2n}\right)!} = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \cdots$

Thus $\cos x$ is expressible in the form of a power series.

From Cosine Function is Absolutely Convergent, we have that the interval of convergence of $\cos x$ is the whole of $\R$.

From Power Series Differentiable on Interval of Convergence, it follows that $\cos x$ is continuous on the whole of $\R$.

$\blacksquare$