The Sum of a Constant
Introduction
What is the sum of the series:
∑ a from x= 0 to n (a is a real or complex constant, n is a positive integer)
I may not be what you think. Take a close look at the limits: lower limit of 0, upper limit of n. Assume the increment of x is 1.
The sum of the series is (n + 1) * a.
Proof
Base case. Let n = 1. Then:
∑ a from x = 0 to 1
= a + a
= 2 * a
= (1 + 1) * a
The value a is added for the x=0 term. The value a is added for the x=1 term.
Induction. Assume for a positive integer k, the series holds. Then for the sum from x = 0 to x = k + 1:
∑ a from x = 0 to k+1
= ( ∑ a from x = 0 to k ) + ( ∑ a from x = k+1 to k+1 )
= (k + 1) * a + a
= k * a + a + a
= k * a + 2 * a
= (k + 2) * a QED
Examples
Example 1:
∑ a from x = 0 to 2
= a + a + a
= 3 * a
Example 2:
∑ 6 from x = 0 to 11
= (11 + 1) * 6
= 12 * 6
= 72
Eddie
All original content copyright, © 2011-2020. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
Introduction
What is the sum of the series:
∑ a from x= 0 to n (a is a real or complex constant, n is a positive integer)
I may not be what you think. Take a close look at the limits: lower limit of 0, upper limit of n. Assume the increment of x is 1.
The sum of the series is (n + 1) * a.
Proof
Base case. Let n = 1. Then:
∑ a from x = 0 to 1
= a + a
= 2 * a
= (1 + 1) * a
The value a is added for the x=0 term. The value a is added for the x=1 term.
Induction. Assume for a positive integer k, the series holds. Then for the sum from x = 0 to x = k + 1:
∑ a from x = 0 to k+1
= ( ∑ a from x = 0 to k ) + ( ∑ a from x = k+1 to k+1 )
= (k + 1) * a + a
= k * a + a + a
= k * a + 2 * a
= (k + 2) * a QED
Examples
Example 1:
∑ a from x = 0 to 2
= a + a + a
= 3 * a
Example 2:
∑ 6 from x = 0 to 11
= (11 + 1) * 6
= 12 * 6
= 72
Eddie
All original content copyright, © 2011-2020. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.