Series Exercises

Answer

(a) The 1st partial sum of a series is its first term. In this case, the 1st partial sum is

2(1) = 2.

(b) The 3rd partial sum is the sum of the first 3 terms:

2(1) + 2(2) + 2(3) = 12.

(c) The 5th partial sum is the sum of the first 5 terms:

2(1) + 2(2) + 2(3) + 2(4) + 2(5) = 30.

We usually denote the nth partial sum of a series by S_n.

Answer

(a) S₂ is the 2nd partial sum, which is the sum of the first 2 terms.

S₂ = 2 + 4 = 6.

(b) S₄ is the 4th partial sum, which is the sum of the first 4 terms.

S₄ = 2 + 4 + 6 + 8 = 20

(c) S₅ is the 5th partial sum, which is the sum of the first 5 terms.

S₅ = 2 + 4 + 6 + 8 + 10 = 30.

Answer

The first four partial sums are

S₁ = 3 = 3

S₂ = 3 + 8 = 11

S₃ = 3 + 8 + 13 = 24

S₄ = 3 + 8 + 13 + 18 = 42

The first 4 terms of the sequence of partial sums are

3, 11, 24, 42.

Answer

The first 4 partial sums are

This means the first 4 terms of the sequence of partial sums are

Answer

Ignore the fact that the indexing starts at 0. If we expand the summation notation, we get

0 + 1 + 4 + 9 + 16 + ...

The first term of the series is 0. We need to account for that when we find the partial sums.

S₁ = 0 = 0

S₂ = 0 + 1 = 1

S₃ = 0 + 1 + 4 = 5

S₄ = 0 + 1 + 4 + 9 = 14

The first 4 terms of the sequence of partial sums are

0, 1, 5, 14

Answer

For any SERIES there is a SEQUENCE of partial sums.

Answer

A partial sum is a SERIES.

Answer

A SERIES is a sum of numbers.

Answer

Either one works. A SEQUENCE can be either finite or infinite. A SERIES can be either finite or infinite. However, remember that we can always think of a series as being infinite.

Answer

A list of numbers separated by commas is called a SEQUENCE.

Answer

We can always think of a SERIES as being infinite. We can't always think of a sequence as being infinite.