**Series**

The word ‘series’ in common language implies much the same thing as ‘sequence’. But in mathematics when we talk of a series, we are referring in particular to sums of terms in a sequence, eg. for a sequence of values an, the corresponding series is the sequence of *S _{n}* with

*S*=

_{n}*a*

_{1}+

*a*

_{2}+⋯+

*a*

_{(n-1)}+

*a*

_{n}If the terms are in an arithmetic sequence, we call the sum an

**arithmetic series**.

**Sigma notation**

Most of the series we consider in mathematics are **infinite** series. This name is used to emphasize the fact that the series contain infinitely many terms. Any sum in the series *S _{k}* will be called a partial sum and is given by

*S*=

_{k}*a*

_{1}+

*a*

_{2}+⋯+

*a*

_{(k-1)}+

*a*

_{k}For convenience, this partial sum is written using the sigma notation:

Sigma notation is a concise and convenient way to represent long sums.

Here, the symbol Σ is the Greek capital letter

__sigma__that refers to the initial letter of the word ‘sum’. So, the expression ∑

^{i=k}

_{i=1}

*a*means the sum of all the terms

_{i}*a*, where

_{i}*i*takes the values from 1 to

*k*. We can also write ∑

^{n}

_{i=m}

*a*to mean the sum of the terms

_{i}*a*, where

_{i}*i*takes the values from

*m*to

*n*. In such a sum,

*m*is called the lower limit and

*n*the upper limit.

Note that we have ∑

^{5}_{n=1}n=∑^{5}_{i=1}i. Thenand theijust play the role of dummy Variables.

We can also work the other way. Sometimes our sum has a pattern which enables us to Write the sum using sigma notation.

Ken wants to get more exercise so he begins by walking for 20 minutes. Each day for two weeks, he increases the length of time that he walks by 5 minutes. At the end of two weeks, the length of time that he has walked each day is given by the following arithmetic sequence:

The total length of time that Ken has walked in two weeks is the sum of the terms of this arithmetic sequence:

This sum is called a series.

Definition

Aseriesis the indicated sum of the terms of a sequence.

The symbol Σ, which is the Greek letter sigma, is used to indicate a sum. The number of minutes that Ken walked on the *n*th day is *a _{n}*=20+(

*n*-1)(5). We can write the sum of the number of minutes that Ken walked in

**sigma notation**:

^{14}

_{n=1}

*a*=∑

_{n}^{14}

_{n=1}20+(

*n*-1)(5)

The “

*n*=1” below Σ is the value of

*n*for the first term of the series, and the number above Σ is the value of

*n*for the last term of the series. The symbol ∑

^{14}

_{n=1}

*a*can be read as “the sum of

_{n}*a*for all integral values of

_{n}*n*from 1 to 14.”

In expanded form:

For example, we can indicate the sum of the first 50 positive even numbers as ∑

^{50}

_{i=1}2

*i*.

^{50}

_{i=1}2i=2(1)+2(2)+2(3)+2(4)+⋯+2(50)

=2+4+6+8+⋯+100

Note that in this case, *i* was used to indicate the number of the term. Although any variable can be used, *n*, *i*, and *k* are the variables most frequently used. Be careful not to confuse the variable *i* with the imaginary number *i*.

The series ∑^{50}_{i=1} 2i is an example of a **finite series** since it is the sum of a __finite__ number of terms. An **infinite series** is the sum of an __infinite__ number of terms of a sequence. We indicate that a series is infinite by using the symbol for infinity, ∞. For example, we can indicate the sum of all of the positive even numbers as:

^{∞}

_{i=1}2i=2+4+6+⋯+2i+⋯

Writing a series below in sigma notation. Notice that the instructions specify that the lower limit of summation should be 1 and the variable *k*. So that would be a good thing to adhere to.

4+11+18+25+ ⋯ for *n* terms It looks arithmetic, with *d*=7. To go in the summand, I’ll need a formula for *u _{n}*. Or, rather, a formula for

*u*, since that’s the index variable.

_{k}*u*=

_{k}*u*

_{1}+(

*k*-1)

*d*=4+(

*k*-1)⋅7

*u*=7

_{k}*k*-3

And now to put that in summation form:

Example 1.

Write the following arithmetic series using sigma notation.

Solution 1:

Write a formula: *u _{n}*=

*u*

_{1}+(

*n*-1)

*d*

List the information:

*u*

_{1}=3,

*d*=10-2=7.

*n*=? (

*n*=6, but leave it as an unknown)

*u*=? (

_{n}*u*=38, but leave it as an unknown)

_{n}Substitute and solve:

*u*=3+(

_{n}*n*-1)⋅7=3+7

*n*-7

*u*=7

_{n}*n*-4

Now let

*n*=i, and write,

*u*=7

_{i}*i*-4.

Hence, 3+10+17+24+31+38=∑

^{6}

_{i=1}(7

*i*-4).