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S3 is the 3rd partial sum, so we need to add the first 3 terms of the series. We get
S3 = 1 + 2 + 4 = 7.
Be Careful: To find Sn, we recommend writing the series in expanded form and ignoring the original indexing.
The original indexing of a series doesn't affect what its partial sums are. Whether a series is indexed starting at 0 or 1 or something else, S1 is the first term we see when we write the series in expanded form.
Find S1 and S2 for the series
In expanded form, the series is
The first partial sum is the first term we see:
S1 = 1.
The next partial sum is the sum of the first two terms:
S2 = 1 + 4 = 5.
Find the sequence of partial sums for the series
1 + 2 + 4 + 8 + 16 + ...
Calculate some partial sums.
S1 = 1 = 1
S2 = 1 + 2 = 3
S3 = 1 + 2 + 4 = 7
S4 = 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 nth term of the sequence is
Sn = 2n – 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 an of the series: a1, a2, a3, ...
The sequence whose terms are the partial sums Sn of the series: S1, S2, S3, ....