Sigma Notation
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First expand the series for each value of n
#n/(2n+1)# =#1/(2(1)+1# +#2/(2(2)+1# +#3/(2(3)+1# +#4/(2(4)+1# +#5/(2(5)+1# Next, perform the operations in the denominator...
#1/(3)# +#2/5# +#3/7# +#4/9# +#5/11# Now, to add fractions we need a common denominator... in this case it's
#3465# Next, we have to multiply each numerator and denominator by the missing components...
#1/3# gets multiplied by#1155# giving#1155/3465# (Divide the
#3465# by#3# to get#1155# and divide the rest by the given denominator.)#2/5*693/693=1386/3465, 3/7*495/495=1485/3465, 4/9*385/385=1540/3465 and 5/11*315/315=1575/3465# Now simply add the numerators together...
#(1155+1386+1485+1540+1575)/3465# giving
#7141/3465# . 
#1/2+1/4+1/8+cdots=1/2^1+1/2^2+1/2^3+cdots=sum_{n=1}^infty1/2^n#
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Introduction to Integration

1Sigma Notation

2Integration: the Area Problem

3Formal Definition of the Definite Integral

4Definite and indefinite integrals

5Integrals of Polynomial functions

6Determining Basic Rates of Change Using Integrals

7Integrals of Trigonometric Functions

8Integrals of Exponential Functions

9Integrals of Rational Functions

10The Fundamental Theorem of Calculus

11Basic Properties of Definite Integrals