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Calculus 1 Problems & Solutions  –  Chapter 6  –  Section 6.1.1

6.1.1
Summation Notation And Formulas

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Review

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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