# At a Glance - Convergence of Series

Now that we know *how* to bake an infinite brownie, we want to know what it tastes like. We are going to need to find a cow big enough to give us a glass of milk to match.

In terms of series, this means adding up infinitely many numbers? We would expect the sum of an infinite list of numbers to be infinite. Some of the time, we'd be wrong.

Before we can understand how we can fit an infinite sum into a finite box, we have to define something called a **partial sum**. A partial sum is what we get if we add up *some* of the terms of a series. This is like breaking off a part of our infinite brownie and sharing it with friends.

More specifically, the ** nth partial sum** of a series is the sum of the first

*n*terms of the series.

### Sample Problem

Find the 4th partial sum of the series

Answer.

To find the 4th partial sum we add the first 4 terms:

The 4th partial sum is .

The 1st partial sum of a series is its first term.

### Sample Problem

The 1st partial sum of the series

is

If you think a series seems like the evil twin of a sequence, you are almost right. A series is the evil *cousin* of the sequence. They are related through a **sequence of partial sums**

*S*_{1}, *S*_{2}, *S*_{3}, *S*_{4},... where the *n*th term is the *n*th partial sum of the series.

**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*of the series:_{n}*a*_{1},*a*_{2},*a*_{3},...

- The sequence whose terms are the partial sums
*S*of the series:_{n}*S*_{1},*S*_{2},*S*_{3},...

These two sequences are *different*.

#### Example 1

Find 1 + 2 + 4 + 8 + 16 + .... |

#### Example 2

Find |

#### Example 3

Find the sequence of partial sums for the series 1 + 2 + 4 + 8 + 16 + ... |

#### Exercise 1

Find the *n*th partial sum of the series

for

(a) *n* = 1

(b) *n* = 3

(c) *n* = 5

#### Exercise 2

Given the series

2 + 4 + 6 + 8 + 10 + ...

find

(a) *S*_{2}

(b) *S*_{4}

(c) *S*_{5}

#### Exercise 3

For the series, find the first 4 terms of the sequence of partial sums.

3 + 8 + 13 + 18 + ...

#### Exercise 4

For the series, find the first 4 terms of the sequence of partial sums.

#### Exercise 5

For the series, find the first 4 terms of the sequence of partial sums.

#### Exercise 6

Fill in the blank with either the word 'sequence' or the word 'series.'

For any BLANK there is a BLANK of partial sums.

#### Exercise 7

Fill in the blank with either the word 'sequence' or the word 'series.'

A partial sum is a BLANK.

#### Exercise 8

Fill in the blank with either the word 'sequence' or the word 'series.'

A BLANK is a sum of numbers.

#### Exercise 9

Fill in the blank with either the word 'sequence' or the word 'series.'

A BLANK can be either finite or infinite.

#### Exercise 10

Fill in the blank with either the word 'sequence' or the word 'series.'

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

#### Exercise 11

Fill in the blank with either the word 'sequence' or the word 'series.'

We can always think of a BLANK as being infinite.