Sigma Notation
Key Questions

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# 
Sigma notation can be a bit daunting, but it's actually rather straightforward. The common way to write sigma notation is as follows:
#sum_(x)^nf(x)# Breaking it down into its parts:
 The
#sum# sign just means "the sum".  The
#x# at the bottom is our starting value for x. It usually has a number next to it:#sum_(x=0)# , for example, means we start at x=0 and carry on upwards until we hit...  The
#n# at the top.  The
#f(x)# is what we need to plug all these values into. At the end, we add the results obtained from here together, and that's our answer.
Note that it's not always
#f(x)#  it is most often#f(n)# or#f(i)# .As an example:
#sum_(x=0)^9(sqrt(x)+1)^2# means we need to find
#(sqrt(0)+1)^2+(sqrt(1)+1)^2+(sqrt(2)+1)^2+...+(sqrt(9)+1)^2# .  The
Questions
Introduction to Integration

Sigma Notation

Integration: the Area Problem

Formal Definition of the Definite Integral

Definite and indefinite integrals

Integrals of Polynomial functions

Determining Basic Rates of Change Using Integrals

Integrals of Trigonometric Functions

Integrals of Exponential Functions

Integrals of Rational Functions

The Fundamental Theorem of Calculus

Basic Properties of Definite Integrals