Calculus Of One Real Variable By Pheng Kim Ving
Chapter 9: The Integral Section 9.1: Summation Notation And Formulas

 

9.1
Summation Notation And Formulas

 

 

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1. Notation

 

 

 

 

Example 1.1

 

Write out these sums:

 

 

Solution


EOS

 

The lower limit of the sum is often 1. It may also be any other non-negative integer, like 0 or 3.

 

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2. Properties

 

 

 

 

Proof

 

ii and iii. Similar to i.

EOP

 

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3. Formulas

 

 

 

 

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

 

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Problems & Solutions

 

1. Write out this sum:

 

 

Solution

 

 

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2. Write out this sum:

 

 

Solution

 

 

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3. Write out this sum:

 

 

Solution

 

 

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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:

 

 

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5.
a. Prove this formula:

 

 

Solution

 

 

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