Calculate some partial sums. *S*_{1} = 1 = 1
*S*_{2} = 1 + 2 = 3
*S*_{3} = 1 + 2 + 4 = 7
*S*_{4} = 1 + 2 + 4 + 8 = 15
The sequence of partial sums is 1, 3, 7, 15, .... By recognizing that these numbers are close to the powers of 2, we can find the general term of the sequence. The *n*th term of the sequence is *S*_{n} = 2^{n} – 1.
**Be Careful:** Remember that sequences and series are different things.
In reality, a series is the evil triplet of two different sequences. Any *series* has *two different sequences* associated with it: - The sequence whose terms are the same as the terms
*a*_{n} of the series:
*a*_{1}, *a*_{2}, *a*_{3}, ... - The sequence whose terms are the partial sums
*S*_{n} of the series:
*S*_{1}, *S*_{2}, *S*_{3}, ....
These two sequences are *different*. | |