Then if W1; W2 are subspaces of V with W1 \ W2 = f0g (the only vector in both of them is 0) and W1 + W2 = V , then we say that V is the (internal) direct sum of W1

Understand how to compute limits of rational functions at infinity. Understand how to use the basic summation formulas and the limit rules you learned in this chapter to evaluate some definite integrals.

There are several ways to summarize data using the SUM function. This paper illustrates various methods ranging from using the SUM function in the simple data step to using the SUM function in …

DEFINITION 5 An in ̄nite sum § n=1(¡1)n+1an; for an ̧ 0 is called ALTERNATING in ̄nite sum (al-ternating series).

Differentiation and integration are two powerful tools for handling various sums that would be impenetrable with other methods. In this handout, we explore some of these methods. This …

Explicit Formulas for Sums Consider the sum of the first n positive integers: X n = + 2 + 3 + · · · + n k=1

Summation notation lets us describe large sums by giving a formula for each term. For example, 12 + 22 + 32 + 42 + 52 + 62 + 72 + 82 + 92 + 102. The idea is that the variable k takes all of the values …