# Precalculus : Sigma Notation

## Example Questions

### Example Question #1 : Sigma Notation

Write out the first 4 partial sums of the following series:

Explanation:

Partial sums (written ) are the first few terms of a sum, so

If you then just take off the last number in that sum you get the  and so on.

### Example Question #1 : Sigma Notation

Express the repeating decimal 0.161616..... as a geometric series in sigma notation.

Explanation:

First break down the decimal into a sum of fractions to see the pattern.

and so on.  Thus,

These fractions can be reduced, and the sum becomes

Each term is multiplied by to get the next term which is added.

For the first 4 terms this would look like

Let  be the index variable in the sum, so if  starts at  the terms in the above sum would look like:

.

The decimal is repeating, so the pattern of addition occurs an infinite number of times.  The sum expressed in sigma notation would then be:

.

### Example Question #3 : Sigma Notation

Evaluate the summation described by the following notation:

Explanation:

In order to evaluate the summation, we must understand what the notation of the expression means:

This sigma notation tells us to sum the values obatined from evaluating the expression at each integer between and including those below and above the sigma. So we're going to start by evaluating the expression at n=1, and then add the value of the expression evaluated at n=2, and so on, until we end by adding the last value of the expression evaluated at n=5. This process is shown mathematically below:

### Example Question #1 : Sigma Notation

Evaluate:

Explanation:

The summation starts at 6 and ends at 7.  Increase the value of  after each iteration:

### Example Question #1 : Sigma Notation

Evaluate:

Explanation:

Rewrite the summation term by term and evaluate.

### Example Question #6 : Sigma Notation

Evaluate:

Explanation:

Rewrite the summation term by term:

To simplfy we get a common denominator of 24.

### Example Question #12 : Sequences And Series

Evaluate:

Explanation:

means add the values for starting with for every integer until .

This will look like:

### Example Question #8 : Sigma Notation

Explanation:

First, evaluate the sum. We can multiply by -2 last.

The sum

means to add together every value for for an integer value of n from 1 to 5:

Now our final step is to multiply by -2.

### Example Question #9 : Sigma Notation

Rewrite this sum using summation notation:

Explanation:

First, we must identify a pattern in this sum. Note that the sum can be rewritten as:

If we want to start our sum at k=1, then the function must be:

so that the first value is .

In order to finish at , the last k value must be 29 because 29-1=28.

Thus, our summation notation is as follows:

Solve: