Why does the blue curve diverge from the red curve in the partial sum approximation?

Conditions

• The blue curve is a finite partial sum (e.g., the first four terms).
• The red curve is the target function $\cos x$.
• $x$ values are sufficiently large (far from the origin).

Reasoning, step by step

1. Identify the blue curve as the polynomial formed by the first four terms: $1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!}$.
2. Observe that for large $x$, the highest-degree term ($-\frac{x^6}{6!}$) dominates the polynomial's value.
3. Note that a negative sixth-degree term forces the polynomial to plunge towards negative infinity as $|x|$ increases.
4. Contrast this with the red curve ($\cos x$), which oscillates periodically between $-1$ and $1$.
5. Conclude that the divergence is an inherent limitation of using a finite number of terms to approximate a periodic function.

Example

The video shows the blue curve (first four terms) following the red cosine curve near the origin but then diverging sharply downwards for larger $x$ values, driven by the $-\frac{x^6}{6!}$ term.

Common misconceptions

• Assuming that adding a few more terms will permanently fix the divergence everywhere.
• Believing that finite polynomials can perfectly replicate periodic behavior over an infinite domain.

Watch the explanation

Explore next

• What is the Maclaurin series for the cosine function?
• How do individual terms affect the shape of the approximation?
• How do partial sums approximate the function defined by the series?