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The blue curve diverges from the red curve because it represents a finite polynomial (a partial sum) rather than the infinite series. Finite polynomials have a dominant highest-degree term that dictates their end behavior, causing them to shoot off to positive or negative infinity as $x$ grows large, whereas the infinite series converges to the bounded, periodic cosine function.

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).

The blue curve diverges from the red curve because it represents a finite polynomial (a partial sum) rather than the infinite series. Finite polynomials have a dominant highest-degree term that dictates their end behavior, causing them to shoot off to positive or negative infinity as $x$ grows large, whereas the infinite series converges to the bounded, periodic cosine function.

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).