CK-12-Calculus

http://www.ck12.org Chapter 4. Integration

Useful Summation Formulas
We can use the notation to indicate useful formulas that we will have occasion to use. For example, you may recall
that the sum of the firstnintegers isn(n+ 1 )/ 2 .We can indicate this formula using sigma notation. The formula is
given here along with two other formulas that will become useful to us.

n

∑i= 1 i=

n(n+ 1 )

2 ,

n

i∑= 1 i^2 =n(n+^1 )(^62 n+^1 ),

n

i∑= 1 i^3 =

[n(n+ 1 )

2

] 2

.

We can show from associative, commutative, and distributive laws for real numbers that
∑ni= 1 (ai+bi) =∑ni= 1 (ai)+∑ni= 1 (bi)and
∑ni= 1 (kai) =k∑ni= 1 (ai).
Example 1:
Compute the following quantity using the summation formulas:

10

∑i= 12 i(i−^6 i).

Solution:

10

i∑= 12 i(i−^6 ) =

10

i∑= 1 (^2 i^2 −^12 i) =^2

10

i∑= 1 i^2 −^12

10

i∑= 1 i

= 2

(( 10 )( 10 + 1 )( 2 · 10 + 1 )

6

)

− 12

(( 10 )( 11 )

2

)

= 770 − 660 = 110.

Another Look at Upper and Lower Sums
We are now ready to formalize our initial ideas about upper and lower sums.
Letfbe a bounded function in a closed interval[a,b]andP= [x 0 ,...,xn]the partition of[a,b]intonsubintervals.
We can then define the lower and upper sums, respectively, over partitionP, by

S(P) =

n

∑ 1 mi(xi−xi−^1 ) =m^1 (x^1 −x^0 )+m^2 (x^2 −x^1 )+...+mn(xn−xn−^1 ),

T(P) =

n

∑ 1 Mi(xi−xi− 1 ) =M 1 (x 1 −x 0 )+M 2 (x 2 −x 1 )+...+Mn(xn−xn− 1 ).

wheremiis the minimum value of fin the interval of lengthxi−xi− 1 andMiis the maximum value off in the
interval of lengthxi−xi− 1.