More on Sequences

Another special sequence and its sum: $T = \langle 1, 4, 9, \dots, n^2\rangle$, the sequence of squares of integers, has the sum $$\sum_{i=1}^n i^2 = 1 + 4 + 9 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}.$$

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• A sequence whose terms always grow (e.g., $\langle 1, 4, 9, \dots, n^2\rangle$,) always diminish (e.g., $\langle 10^n, \dots, 10^2, 10^1, 10^0\rangle$,) or are always constant is called a monotone sequence.
• A sequence whose terms are always constant (e.g., $\langle 7, 7, 7, \dots, 7\rangle$) is called a constant sequence.

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Fun class activities: (1) What is the sum $\sum_{i=1}^{15} i$?   (2) The sum $\sum_{i=1}^5 i^2$?   (3) $5!$   (4) The sum $\sum_{i=0}^\infty \frac{1}{2^i}$?