Sigma Notation

1. We can write u1 + u2 + u3 + u4 + ... + un using sigma notation.
2. Σ is called sigma which is the equivalence of capital S in the Greek alphabet. The first use of Σ as a notation for summation was attributed to Leonhard Euler. [Euler was born in Basle on April 15, 1707. He was widely aknowledged as "Analysis Incarnate."]
3.  n Σ ur = u1 + u2 + u3 + u4 + ... + un r =1
4.  4 Σ (3k+5) = [3(1)+5] + [3(2)+5] + [3(3)+5] + [3(4)+5] k =1

 4 Σ (3k+5) = [8] + [11] + [14] + [17] = 50 k =1
Note that this is actually an arithmetic series or summation.
5.  6 Σ (1/3)i = 1/3 + 1/9 + 1/27 + 1/81 + 1/243 + 1/729 i =1
which has a sum of 364/729. Note that this is actually a geometric series or summation. Note that the subcript can be written as r, k or i.
6. Some Properties of Sigma Notation.
 n n n Σ [ ur + vr ] = Σ ur + Σ vr r =1 r =1 r =1

 n n Σ k ur = k Σ ur r =1 r =1
where k is some constant term.
 n Σ k = kn r =1
This is equivalent to adding a constant k, n number of times.
7.  ∞ Σ ur r =1
means summing ur from r=1 to infinity.

Exercises

1. Expand the following:
1.  5 Σ (2 + 1/r) r = 1
2.  4 Σ k(3-k) k = 1
3.  6 Σ 5(2)i -1 i = 1
2. Find the sum of these series:
1.  200 Σ (3r-5) r = 1
2.  10 Σ 7(1.3)k-2 k = 1
3.  10 Σ (2/3)r r = 5