11.3 The Integral Test and Estimates of Sums
In general, it is difficult to find the exact sum of a series.
We were able to accomplish this for geometric series and
the series 1/[n(n + 1)] because in
11.6 Absolute Convergence and Ratio and Root Tests
Given any series an, we can consider the corresponding
series
|an | = |a1| + |a2| + |a3| + . . .
whose terms are the absolute values of the terms of
11.9 Representations of Functions as Power Series
We start with an equation that we have seen before:
We have obtained this equation by observing that the series
is a geometric series with a = 1 and r
11.7 Strategy for Testing Series
We now have several ways of testing a series for
convergence or divergence; the problem is to decide which
test to use on which series. In this respect, testing series
9.5
Linear Equations
A first-order linear differential equation is one that can be
put into the form
where P and Q are continuous functions on a given interval.
This type of equation occurs frequently
WESTERN UNIVERSITY
Department of Applied Mathematics
Calculus 1301B - Winter 2017, Section 06
Instructor
Dr. Zinovi Krougly, Office MC 250
Phone 519 661 -2111 ext. 88787
Email [email protected]
We
7.8 Improper Integrals
In this section we extend the concept of a definite integral to
the case where the interval is infinite and also to the case
where f has an infinite discontinuity in [a, b].
In
7.4 Integration of Rational Functions by Partial Fractions
In this section we show how to integrate any rational
function (a ratio of polynomials) by expressing it as a sum of
simpler fractions, calle
10.4 Areas and Lengths in Polar Coordinates
In this section we develop the formula for the area of a
region whose boundary is given by a polar equation. We
need to use the formula for the area of a se
11.2
Series
What do we mean when we express a number as an infinite
decimal? For instance, what does it mean to write
= 3.14159 26535 89793 23846 26433 83279 50288 . . .
The convention behind our dec
Polar Coordinates
A coordinate system represents a point in the plane by an
ordered pair of numbers called coordinates. Usually we use
Cartesian coordinates, which are directed distances from
two perp
10. 2 Calculus with Parametric Curves
Tangents
Suppose f and g are differentiable functions and we want to
find the tangent line at a point on the curve where y is also a
differentiable function of x.
11.1
Sequences
A sequence can be thought of as a list of numbers written in
a definite order:
a1, a2, a3, a4, . . . , an, . . .
The number a1 is called the first term, a2 is the second term,
and in ge
11.8
Power Series
A power series is a series of the form
where x is a variable and the cns are constants called the
coefficients of the series.
For each fixed x, the series
is a series of constants th
10.1 Curves Defined by Parametric Equations
Imagine that a particle moves along the curve C shown in
Figure 1. It is impossible to describe C by an equation of the
form y = f(x) because C fails the Ve
11.5
Alternating Series
In this section we learn how to deal with series whose terms
are not necessarily positive. Of particular importance are
alternating series, whose terms alternate in sign.
An al
11.10 Taylor and Maclaurin Series
We start by supposing that f is any function that can be
represented by a power series
f (x) = c0 + c1(x a) + c2(x a)2 + c3(x a)3 + c4(x a)4
+ .| x a | < R
Lets try t
11.4
Comparison Tests
In the comparison tests the idea is to compare a given
series with a series that is known to be convergent or
divergent. For instance, the series
reminds us of the series
, which
7.3 Trigonometric Substitution
In finding the area of a circle or an ellipse, an integral of the
form
dx arises, where a > 0.
If it were
the substitution u = a2 x2 would be
effective but, as it stands
7.5 Strategy for Integration
In this section we present a collection of miscellaneous
integrals in random order and the main challenge is to
recognize which technique or formula to use.
No hard and fa
9.3 Separable Equations
A separable equation is a first-order differential equation in
which the expression for dy/dx can be factored as a function
of x times a function of y.
In other words, it can b