What is the Maclaurin series for the cosine function?

Conditions

• The variable $x$ is a real number.
• The series is evaluated at $x=0$ (Maclaurin series).

Reasoning, step by step

1. Identify the expanded series $1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots$ from the video.
2. Recognize the pattern of alternating signs, even powers of $x$, and even factorials.
3. Write the general summation form $\sum_{k=0}^{\infty}\frac{(-1)^k x^{2k}}{(2k)!}$.
4. Note the video's conclusion that this infinite series maps exactly onto the graph of $\cos x$.

Example

At the end of the clip, the label '$\cos x$' is written above the series, formally identifying the infinite power series as the Maclaurin representation of the cosine function.

Common misconceptions

• Confusing the Maclaurin series for cosine with the series for sine, which uses odd powers.
• Assuming the series only equals $\cos x$ for small values of $x$, rather than for all real $x$.

Watch the explanation

Explore next

• How do partial sums approximate the function defined by the series?
• Why does the blue curve diverge from the red curve?
• How do individual terms affect the shape of the approximation?