Why does the video say a function can be defined by a series?

Partial sums approximate the function by truncating the infinite series after a finite number of terms to create a polynomial. As more terms are added, the polynomial matches the target function's graph more closely over a wider interval around the origin.

Conditions: The approximation is evaluated near the center of expansion ($x=0$).; A finite number of terms is selected from the infinite series.

The trailing ellipsis indicates that the series continues indefinitely following the exact same generating rule established by the first few terms. It means you keep substituting successive integers for $k$ into the formula $\frac{(-1)^k x^{2k}}{(2k)!}$ to produce the next terms, alternating signs and increasing the even powers and factorials.

Conditions: The first few terms have been explicitly calculated to establish a pattern.; The summation is an infinite series.

To expand the summation, substitute successive integer values for the index $k$ into the general term. For $k=0$, the term simplifies to $1$.

Conditions: The summation index $k$ starts at $0$ and increases by $1$ for each term.; The general term is $\frac{(-1)^k x^{2k}}{(2k)!}$.

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 $k=0$ term equals $1$ because substituting $k=0$ into the general term yields $\frac{(-1)^0 x^0}{0!}$. The video uses the standard conventions that $(-1)^0 = 1$, $x^0 = 1$, and $0! = 1$.

Conditions: The index value is $k=0$.; The standard mathematical conventions $x^0=1$ and $0!=1$ are assumed.