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

 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.

 2. Properties

Proof

ii and iii.  Similar to i.

EOP

 3. Formulas

Proof
i.  We use the identity (k + 1)2k2 = 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)3k3 = 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)4k4 = 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