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CALCULUS 1 PROBLEMS & SOLUTIONS

6.1.1
Summation Notation And Formulas


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Review


1. Notation



The symbol on the left-hand side is read: "the sum of the a_i as i runs from 1 to n", or, for short,
“ sigma a_i i from 1 to n”. The letter S stands for sum. The letter i is called the summation index. The
integer 1 below the letter S is where the sum starts, and is called the lower limit of the sum. The
integer n above the letter S is where the sum stops, and is called the upper limit of the sum.


Examples 1



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2. Properties



Proof


The proof of each of the other two identities is similar.
EOP

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3. Formulas



Proof Of First Formula

EOP

Proof Of Second Formula

EOP

The proofs of the third and fourth formulas are left as Problems & Solutions 2 and 3 respectively.

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Problems & Solutions


1. Write out the following sums.



Solution



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2. Prove this formula:



Solution



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3. Prove this formula:



where r ¹ 1.

Solution



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