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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.

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.