What does the trailing ellipsis mean in the expanded series $1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\ldots$?

Conditions

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

Reasoning, step by step

1. Observe the pattern of the first four terms: alternating signs, even powers of $x$, and even factorials in the denominator.
2. Recognize that the ellipsis is not random decoration but a mathematical instruction.
3. Apply the general term formula $\frac{(-1)^k x^{2k}}{(2k)!}$ for $k=4, 5, 6, \ldots$ to generate subsequent terms.
4. Conclude that the series is infinite and follows the established rule.

Example

The video boxes the entire expansion and the speaker notes that the series "goes on," emphasizing that the ellipsis represents the continuation of the same term-generating rule.

Common misconceptions

• Treating the ellipsis as meaningless decoration.
• Assuming the pattern stops or changes arbitrarily after the written terms.
• Thinking the ellipsis implies a finite number of unknown terms rather than an infinite continuation.

Watch the explanation

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