510
Chapter 5 | Integration
5.
∑ i =1 n
a i = ∑ i =1 m
n
(5.5)
a i + ∑
a i
i = m +1
Proof We prove properties 2. and 3. here, and leave proof of the other properties to the Exercises. 2. We have ∑ i =1 n ca i = ca 1 + ca 2 + ca 3 +⋯+ ca n = c ( a 1 + a 2 + a 3 +⋯+ a n ) = c ∑ i =1 n a i . 3. We have ∑ i =1 n ⎛ ⎝ a i + b i ⎞ ⎠ = ⎛ ⎝ a 1 + b 1 ⎞ ⎠ + ⎛ ⎝ a 2 + b 2 ⎞ ⎠ + ⎛ ⎝ a 3 + b 3 ⎞ ⎠ +⋯+ ⎛ ⎝ a n + b n
⎞ ⎠
= ( a 1 + a 2 + a 3 +⋯+ a n )+ ⎛
⎝ b 1 + b 2 + b 3 +⋯+ b n ⎞ ⎠
= ∑ i =1 n
a i + ∑ i =1 n
b i .
□ A few more formulas for frequently found functions simplify the summation process further. These are shown in the next rule, for sums and powers of integers , and we use them in the next set of examples.
Rule: Sums and Powers of Integers 1. The sum of n integers is given by ∑ i =1 n
i =1+2+⋯+ n = n ( n +1) 2 .
2. The sum of consecutive integers squared is given by ∑ i =1 n 3. The sum of consecutive integers cubed is given by ∑ i =1 n
i 2 =1 2 +2 2 +⋯+ n 2 = n ( n +1)(2 n +1) 6 .
2 ( n +1) 2
i 3 =1 3 +2 3 +⋯+ n 3 = n
4 .
Example 5.2 Evaluation Using Sigma Notation
Write using sigma notation and evaluate: a. The sum of the terms ( i −3) 2 for i = 1, 2,…, 200.
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