Alternating Series

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Absolute Convergence

If a series has some positive and some negative terms, there are a couple of things that one might ask. The first is
1) does the series converge? Another question, the motivation for which is less obvious, is
2) does the series converge if we take the absolute values of its terms?

If the first answer is yes, the second can be yes or no. It turns out that if this second answer is yes, the series behaves much like a finite sum, i.e. it behaves well.

Definition: Let $\sum a_n$ be a series.

If the series $ {\sum \left|a_n\right|}$ converges, then we say that $\sum a_n$ is absolutely convergent.

If $\sum a_n$ converges but $\sum \left|a_n\right|$ doesn't, then we say that $\sum a_n$ is conditionally convergent.

Example: Consider the alternating harmonic series $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = 1 - \frac12 + \frac13 - \frac14+\cdots.$$It converges (we saw this previously by using the AST). The series with the absolute values of its terms, which is the harmonic series $\sum \frac{1}{n}$, diverges ($p$-series with $p\le 1$). Since the series converges, but not in absolute value, we say it is conditionally convergent.

One fact, said in two ways

If $\sum \left|a_n\right|$ converges, then $\sum a_n$ converges.
(Absolutely convergent $\Longrightarrow$ convergent.)

If $\sum a_n$ does not converge, then $\sum\left|a_n\right|$ will not converge.
(Divergent $\Longrightarrow$ not absolutely convergent.)

Example: Consider the alternating $p$-series, with $p=2$, $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^2} = 1 - \frac14 + \frac19 - \frac1{16}+\cdots.$$Since the series with the absolute values of the terms of our series, $\sum\frac{1}{n^2}$, is a convergent $p$-series, our series is absolutely convergent. By the fact above, this means it is also convergent. It is not conditionally convergent.

Be careful with these terms

Conditional convergence of a series means it is convergent but not absolutely convergent.

If we are told that a series is convergent, we do not know a priori whether it is conditionally convergent or absolutely convergent. It is one or the other, but not both.

Every series is either divergent, conditionally convergent, or absolutely convergent, but it is only one of these things.

Justification of the fact above, and some examples, are discussed in the video.

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Modeling with Differential Equations

Euler's Method and Direction Fields

Separable Equations

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Linear Equations

Parametrized Curves

Calculus with Parametrized Curves

Polar Coordinates

Areas and Lengths of Polar Curves

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Conic Sections in Polar Coordinates

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Infinite Series

Integral Test

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Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series

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Functions of 2 and 3 variables

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Differentiability and the Chain Rule

Multiple Integrals

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Double Integrals over General Regions

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Multiple integrals in physics

Integrals in Probability and Statistics

Change of Variables

Absolute Convergence

Be careful with these terms