MAT 21B Lecture Notes - Lecture 3: Summation, Royal Institute Of Technology, Partial Fraction Decomposition
§5.2: Sigma Notation and Limits of Finite Sums
The sigma notation P
Often, we need to write sums with a large number of terms
a1+a2+· · · +an.
We may write it in a compact form using sigma notation:
n
X
k=1
ak=a1+a2+· · · +an.
Here,
•Σ: the summation symbol, stands for “sum” (Greek letter sigma)
•k: the index of summation. It tells us where the sum begins (at the number below the Psymbol) and
where it ends (at the number above P).
•ak: a formula for the kth term.
Example What do the following sigma notations stand for?
•P5
k=3 sin kπ
2
•P5
k=0
k
k+1
•P100
m=1
(−1)m
m
Any letter can be used to denote the index, but typically we use i,j, k or m, n. The index may start at
any integers, but often it starts with 1 or 0. Also note that signs are alternating here!
•P5
k=1 3
In general, n
X
k=1
c=nc.
Express the following sums in sigma notation.
•13+ 23+ 33+ 43+ 53+ 63+ 73
•f(2) + f(3) + f(4) + · · · +f(234)
•3+5+7+9+11
These are sums of odd numbers. It may be written as
•P6
k=2 (2k−1)
•or P5
i=1 (2i+ 1)
•or P4
j=0 (2j+ 3)
The same sum may be expressed in different forms!
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Document Summary
5. 2: sigma notation and limits of finite sums. Often, we need to write sums with a large number of terms a1 + a2 + + an. We may write it in a compact form using sigma notation: Here, ak = a1 + a2 + + an. n. Xk=1: : the summation symbol, stands for sum (greek letter sigma, k : the index of summation. It tells us where the sum begins (at the number below the p symbol) and where it ends (at the number above p): ak : a formula for the kth term. Any letter can be used to denote the index, but typically we use i,j, k or m, n. the index may start at any integers, but often it starts with 1 or 0. Also note that signs are alternating here! k=3 sin(cid:0) k . 2 (cid:1) k k=0 k+1 ( 1)m m m=1: p5, p5, p100, p5 k=1 3.
