MATH 118, LECTURES 16, 17 & 18: SEQUENCES
1 Sequences
Many processes in mathematics and the applied sciences result in sequences of numbers. Consider many of the processes of approximation we have carried out in this and the previous semesters Calcu

MATH 118, LECTUREs 19, 20 & 21: SERIES: INTEGRAL & COMPARISON TESTS
1 Series
We are frequently interested in taking the sum of an innite number of elements, that is to say, something of the form
n=1
cn = c1 + c2 + c3 + + cn +
where the ci

MATH 118, LECTURES 22 & 23: SERIES: RATIO & ROOT TESTS
1
1.1
Series
Limit Ratio Test
Consider the convergence of the series
n=1
1 . n!
This can be handled by the comparison test, taking an appropriate choice of bounding function. We will, howev

MATH 118, LECTURE 24: ABSOLUTE CONVERGENCE
1 Absolute Convergence
All of the series n=1 cn we have dealt with so far have had nonnegative terms cn . This begs the question of how we determine the convergence/divergence of series with an innite numb

MATH 118, LECTURE 25: VALUES OF INFINITE SERIES
1 Values of Innite Series
We have done a signicant amount of work in determining whether a series converges or not but have not given any consideration to which value the series converges to (if it doe

MATH 118, LECTURE 26: POWER SERIES
1 Power Series
A few weeks ago when considering the convergence of geometric series we encountered the example 2 n n1 x . 3
n=1
We considered the convergence as we did for normal numerical geometric series and fo

MATH 118, LECTURES 27 & 28: TAYLOR SERIES
1 Taylor Series
Suppose we know that the power series an (x c)n
n=0
converges on some interval c R < x < c + R to the function f (x). That is to say, we have f (x) = a0 + a1 (x c) + a2 (x c)2 + a3 (x

MATH 118, LECTURE 29: BINOMIAL EXPANSION
1 BINOMIAL EXPANSION
We know by binomial expansion that for nonnegative integers m = 0, 1, 2, 3, . . . we have m m n mn (a + b)m = ab n
n=0
where we have the m choose n operator given by m n = m! . (m n)! n

MATH 118, LECTURE 31: TAYLOR REMAINDERS
1 Taylor Remainders
In applications where we are required to use the Taylor series expansion of a function, we are not able to compute all of the terms in the series since there are innitely many. As we did wi

MATH 118, LECTURES 14 & 15: POLAR AREAS
1 Polar Areas
We recall from Cartesian coordinates that we could calculate the area under the curve by taking Riemann sums. We divided the region into subregions, approximated the area over that subregion by a

MATH 118, LECTURES 13 & 14: POLAR EQUATIONS
1 Polar Equations
We now know how to equate Cartesian coordinates with polar coordinates, so that we can represents points in either form and understand what we are talking about. Furthermore, we know how

MATH 118, LECTURE 1: Review & Integration by Substitution
1 Course Information
For a detailed breakdown of the course content and available resources, see the Course Syllabus (general course folder). Other relevant information for this section of MA

MATH 118, LECTURE 2: Integration by Parts
1 Integration By Parts
Consider trying to take the integral of xex dx. We could try to nd a substitution but would quickly grow frustrated there is no substitution we can make which would simplify the integ

MATH 118, LECTURE 3: Trigonometric Integrals
1 Trigonometric Integrals
Many trigonometric integrals can be solved with simple manipulation. We will break the problems for this lecture into three sections: integrals involving sin(x) and cos(x), int

MATH 118, LECTURES 4 & 5: Trigonometric Substitutions
1 Trigonometric Substitution
In this lecture, we are going to use some of our trigonometric identities to simplify integrals. The identities we use relate to the form of derivatives we know: the

MATH 118, LECTURES 5 & 6: COMPLETING THE SQUARE
1 Completing the Square
So far we have learned how to handle integrals with trouble terms of the form a2 b2 x2 , a2 +b2 x2 , and b2 x2 a2 by using trigonometric substitutions. The common thread with th

MATH 118, LECTURES 7 & 8: PARTIAL FRACTIONS
1 Partial Fractions
Consider being asked to solve x dx. (x 1)(x + 1)(x + 3) We may be tempted to try a variety of dierent substitutions; however, there is a signicantly simpler method by which this integr

MATH 118, LECTURES 9: NUMERICAL INTEGRATION
1 Numerical Integration
In evaluating denite integrals, we required that we were able to nd the antiderivative of the function, but consider being asked to evaluate
b a
ex dx.
2
2
There is no closed-for

MATH 118, LECTURES 11: PARAMETRIC EQUATIONS
1 Parametric Equations
We are used to seeing functions written in the form of, for example, y = f (x) which expresses that y depends explicitly on x. When each x determines a unique value of y (verticle li

MATH 118, LECTURE 12: POLAR COORDINATES
1 Polar Coordinates
For many applications, the dependence of y on x (y = f (x) is not the most practical approach to modelling a problem. For instance, imagine you are manning a lighthouse late at night when y

MATH 118, LECTURES 32 & 33: APPLICATIONS OF TAYLOR SERIES
1 Applications of Taylor Series
We have seen how Taylor series can provide a useful approximation of a function on an interval. Furthermore, we have been able to use Taylors remainder theorem