Calculus II
Contents
Series
Integrals
Definite integrals
Some primitives
Numerical methods
Exercices
© The scientific sentence. 2010
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Calculus II:
Definite integral
Sequences ans series
Convergence and divergence tests in Series
We denote by {an} the sequence of general term an and
by {Sn} the nth partial sum of the series Σ 1 ∞
Strategy for Testing Series
We have several ways of testing a series for convergence or divergence, but the problem
is to decide which test to use on which series.
The testing series is similar to integrating functions. We have no fixed rules
to apply to series to test their convergence. But there are some convenient ways
about which test to apply to a given series.
The bad strategy is to apply a list of the tests in a specific
order until one finally works. That would be a waste of time and effort.
Instead, as with integration, the main strategy is to
classify the series according to its form.
1. If
limn → ∞ Sn ≠ 0, then the Test for Divergence
should be used.
2. If the series is of the form Σ{1/np}, it is a p-series, which we know to be convergent if p > 1 and divergent if p ≤ 1 .
3. If the series has the form Σ{arn} or , Σ{arn - 1, it is a geometric series, which converges if |r| < 1 and diverges if |r| ≥ 1.
Some preliminary algebraic manipulation
may be required to bring the series into this form.
4. If the series has a form that is similar to a p-series or a geometric series, then
one of the comparison tests should be considered.
In particular, if Sn is a rational function or
algebraic function of n (involving roots of polynomials), then the
series should be compared with a p-series.
The value of p should be chosen by keeping only the highest powers of n
in the numerator and denominator.
The comparison tests apply only to series with positive terms, but
if Σ{Sn} has some negative terms, then we can apply the
Comparison Test to Σ{|Sn|}
and test for absolute convergence.
5. If the series is of the form Σ{(-1)n-1 bn} or Σ{(-1)n bn, then the Alternating Series Test is an obvious possibility.
6. Series that involve factorials or other products (including a constant raised to the
nth power) are often conveniently tested using the Ratio Test.
Bear in mind that |Sn+1/Sn| →1 as
n → ∞ for all p-series and therefore all rational or algebraic
functions of n. Thus, the Ratio Test should not be used for such series.
7. If Sn = f(n), where ∫1∞ f(x) dx is easily
evaluated, then the Integral Test is effective.
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