9.1 Summation Notation And Formulas
| Calculus Of One Real Variable – By Pheng Kim Ving |
| 9.1 |
Return To Contents
Go To Problems & Solutions
Example 1.1
Write out these sums:
EOS
The lower limit of the sum is often 1. It may also be any other non-negative integer, like 0 or 3.
Go To Problems & Solutions Return To Top Of Page
Proof
ii and iii. Similar to i.
EOP
Go To Problems & Solutions Return To Top Of Page
Proof
i. We use the identity (k + 1)2 – k2 = 2k + 1 (derived from (k + 1)2 = k2 + 2k + 1). Writing it out for each integer k from
1 to n and adding them up we get:
ii. We use the identity (k + 1)3 – k3 = 3k2 + 3k + 1 (derived from (k + 1)3 = k3 + 3k2 + 3k + 1). ). Writing it out for each
integer k from 1 to n and adding them up we get:
iii and iv. Left as Problems & Solutions 4 and 5 respectively.
EOP
Remark 3.1
For formulas i, ii, and iii, the base is increasing from 1 to n and the exponent is fixed, for example 12 + 22 + … + n2, while
for formula iv the base is fixed and the exponent is increasing from 0 to n, for example 1 + (1/2) + (1/2)2 + … + (1/2)n.
Example 3.1
Find the following sums.
b. 12 + 22 + … + 1002.
Solution
EOS
Return To Top Of Page
1. Write out this sum:
Solution
Return To Top Of Page
2. Write out this sum:
Solution
Return To Top Of Page
3. Write out this sum:
Solution
Return To Top Of Page
4.
a. Prove this formula:
Solution
a. Writing the identity (k + 1)4 – k4 = 4k3 +6k2 + 4k + 1 for each integer k from 1 to n and adding them up we get:
Return To Top Of Page
5.
a. Prove this formula:
Solution
Return To Top Of Page Return To Contents