Summations

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Sunday, 6:00 - 7:30. Review material from Wednesday or Thursday to today, plus time for problem set questions or other review.

Summations

Based on “Sigma (Summation) Notation” in section 5.1 of the textbook and this discussion of summations.

Reading the Notation

What is the sum from i = 1 to 3 of 2i - 1?

Sigma notation provides an expression involving some variable and a range of values for that variable, and represents adding up all the values of the expression for each value of the variable:

The number above the “Σ” is the highest value of the variable, not the number of terms in the sum (sometimes those quantities are the same, but that’s coincidence). For example this sum has an upper bound of 3 but only 2 terms:

Writing the Notation

Use Sigma notation to write the sum 3+4+5+6+...+99+100.

There are several ways to write this, depending on how you want to accommodate the fact that the sum starts at 3:

Both of the sums above are correct.

Evaluating Sums

Use algebra and summation laws to find the sum from i = 1 to 100 of i.

This is a classic example of using a “closed form” formula for a summation, i.e., a formula that gives a value equal to the sum’s without explicitly adding all the numbers together. You can find tables of such formulas (for example, in our textbook). In this case, the relevant closed form formula says that the sum from 1 to n of i is equal to (n/2)(n+1).

How about the sum from j = 1 to 10 of 3j² + j/2?

This example uses a lot of algebraic rules about summations.

First, summations obey “sum” and “difference” rules, due to the fact that (f(1)+g(1)) + (f(2)+g(2)) + ... is the same as (f(1) + f(2) + ...) + (g(1) + g(2) + ...), or, in more formal terms, addition is associative:

Then there’s a “constant multiple” rule, due to the fact that common factors can be factored out of sums:

Now the sums are simplified to the point where there are closed form formulas that apply:

Finally, do the remaining arithmetic to get a final numeric answer:

The upper bound for a summation is often a variable or expression, and in fact that’s the form we’ll see most sums in later in this course. You can use the same algebra and closed form formulas on such summations as you would on ones with constant upper bounds. For example, the sum from i = 1 to n-1 of 3i - 2 is...

Next

Use summation notation to define equations for sums of areas under graphs.

Please read “Approximating Area” and “Forming Riemann Sums” in section 5.1 of the textbook by class time Monday.

Please also contribute to this discussion of areas under curves.

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