Further Concepts for Advanced Mathematics - FP1

Further Concepts for Advanced Mathematics - FP1

Unit 5 Induction & Series – Section5b Summation of Finite Series

The Sum of a Finite Series

Sigma notation can be used to write the sum of a series in a compact way.

For example:

The series can be written as

This series does not converge but provided there are a finite number of terms, the sum of the series can be found. For , the sum is 36.

There are several methods for finding sums of finite series. The two that are used for FP1 are the method of differences and using standard results.

The Method of Differences

This method needs the expression to be written as a subtraction.

For the example above, can be rewritten as the difference

If you multiply out the brackets and simplify the expression, you will find this to be true.

On the whole, you are not expected to find the difference formula yourself (unless it is easy or obvious). You will usually be given the equivalent expression and asked to confirm that it is correct.

Once the difference formula has been confirmed, the value of the sum can be found by substituting successive values of into the formula and tabulating the results.

So the sum of the series which is can be found using the difference formula .

The table for this would look like this:

You can see from the table that some values will cancel each other out from one row to the next.

The cancelling looks like this:

After all of the cancelling has been done, there are only two terms

left. One is 0 and the other is .

Hence,

This method will only work if the original term expression can be rearranged to a difference that results in the sort of cancelling seen above.

Examples

1. i) Show that

ii) Hence find (this is )

Answer

i) Working with the left hand side of the expression gives

which is the same as the right hand side

ii) Putting the values into a table gives

Hence,

In this example, the rearrangement of into could have been found by splitting into partial fractions. This method is not part of FP1 but it is quite straightforward and you could find out how to do it with little effort.

2. i) Show that

ii) Hence find

This example illustrates the point that the cancelling is not always obvious and that a general formula for the sum in terms of can also be found by this method.

i)

ii) The table for this is on the next page

The cancelling here works across three terms diagonally. This will leave three terms in an ‘L’ shape in the bottom right hand corner of the table. To get all of the terms necessary, the term before has to be included. This is .

Hence,

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