Summation notation
Summation notation is a convenient way to represent the sum of many terms. (It iis also called Sigma notation, since it uses the Greek letter Σ (Sigma)). If f is a function, then b f (k) = f (a) + f (a + 1) + f (a + 2) + . . . + f (b). k=a . Here are some examples: • • • •
5 k=1

k 3 = 13 + 23 + 33 + 43 + 53 = 225. = 1/1 + 1/2 + 1/3 + . . . + 1/100 ≈ 5.1873. = 1/1 + 1/2 + 1/3 + . . . + 1/n.

100 1 k=1 k n 1 t=1 t 15 x=1

2 = 2 + 2 + . . . + 2 = 15 · 2 = 30 15 times

•

8 x=2

sin(π/x) = sin(π/2) + sin(π/3) + . . . + sin(π/8) ≈ 4.4885

Here are some handy rules for working with sums, where c is some constant: • • • n x=m

cf (x) = c

n x=m

f (x). n x=m n x=m

n x=m (f (x) n x=m (f (x)

− g(x)) = + g(x)) =

f (x) − f (x) +

n x=m n x=m

g(x). g(x).

Here are some classical formulas that might come in handy: • • • • • n i=1 n i=1 n i=1

1 = n. c = cn. i= n(n+1) . 2 n(n+1)(2n+1) . 6 n(n+1) 2 2

n 2 i=1 i n 3 i=1 i

= =

.

For details, see Appendix E in the text. Most of the second set of formulas can be proved using mathematical induction. 256 2π Calculator sums. It is often useful to use the calculator to find sums such as k=1 cos2 π + k 256 . This would be tedious to calculate by hand. The two commands to use are sum and seq, which on the TI-83, can be found in the LIST menu, under the OPS and MATH sub-menus, respectively. The command seq has the format: seq(expression,variable, lower, upper) This returns a list. For example, seq(X2 , X, 5, 12) returns the list {52 , 62 , . . . 122 }. Given a list, the command sum will add up the terms. For example, sum({5, 4, 9, 2, 10, 8}) will return 5 + 4 + 9 + 2 + 8 = 28. 256 2π We combine these functions to calculate sums. For example, to calculate k=1 cos2 π + k 256 we issue the command sum(seq((cos(π+X*(2π/256)))2 ,X,1,256))) It will take a minute or two to get our answer, 128. By the way, the variable k is a dummy-variable—it doesn’t matter that it is called k. I make my life easier by using x instead (since there is an x key). In general, the syntax for our command for getting should be sum(seq(expression, variable, lower, upper)) Here are some more examples: 1

• • •

= 1/1 + 1/2 + 1/3 + . . . + 1/100: sum(seq(1/X, X, 1, 100)) k 3 = 13 + 23 + 33 + 43 + 53 : sum(seq(X3 , X, 1, 5)) sin(π/x) = sin(π/2) + sin(π/3) + . . . + sin(π/8): sum(seq(sin(π/X),X, 2, 8))
8 x=2 5 k=1

100 1 k=1 k

On the TI-89, there is a better way to do sums. The command is Σ(, which is under the Calc menu. The syntax is similar: Σ(expression, variable, lower, upper) 100 1 So, to evaluate k=1 k = 1/1 + 1/2 + 1/3 + . . . + 1/100, you type in Σ(1/X, X, 1, 100)). It calculates the value exactly, giving us a large fraction. (It also will pretty-print our input, so we know we are doing it correctly.) To get an approximation, you can put in a decimal point somewhere in the expression: Σ(1./X, X, 1, 100)). This gives us 5.187333518. You can also use the sum and seq commands as on the TI-83.

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