Convergence of Series Examples

Find S₃ for the series

1 + 2 + 4 + 8 + 16 + ....

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 S_n, 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

S_n = 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 a_n of the series:
  a₁, a₂, a₃, ...
• The sequence whose terms are the partial sums S_n of the series:
  S₁, S₂, S₃, ....

These two sequences are different.