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< < | ## Infinite Series: A Synopsis | |||||||

> > | ## Infinite Series: A Synopsis | |||||||

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< < | ## Famous Series## Geometric Series- if converges with sum
- if diverges
## P-Series- for diverges
## Harmonic Series- diverges
- this is a special case of the P-Series for P=1
## Alternating Harmonic Series- converges to
*ln(1+1) = ln(2)*using series for*ln(1+x)*below.
## Exponential- converges
where ## Sin- converges
## Cos- converges
##
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> > | ## Famous Series
## Geometric Series
## P-Series
## Harmonic Series
- diverges
- this is a special case of the P-Series for P=1
## Alternating Harmonic Series
- converges to
*ln(1+1) = ln(2)*using series for*ln(1+x)*below.
## Exponential
where ## Sin
## Cos
##
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< < |
<--/twistyPlugin-->
## Useful Limits | |||||||

> > | ## Useful Limits | |||||||

## Exponentials
## Powers and Roots
## Properties of Limits## The limit of a sum is the sum of the limits:
## The limit of a product is the product of the limits:
## The limit of a quotient is the quotient of the limits:
## Squeeze Theorem:
## AKA the "Flyswatter Theorem". Why?
## Ratios of PolynomialsThe limiting behavior of the ratio of two polynomials depends on the degree of the polynomials. The polynomial of higher degree "wins". If their degrees are the same, then the limit is the ratio of the leading coefficients.
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< < | ## Useful Inequalities## Always | |||||||

> > | ## Useful Inequalities
## Always
## Eventually
- for any
- (!) Remember this when using the
*Test for Divergence*
## New Series From Old
## Multiply by a constant:- Before
- After
## Substitute an expression for
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< < |
## Eventually
- for any
- (!) Remember this when using the
*Test for Divergence*
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> > | ## Convergence Tests%INCLUDE{ConvergenceTests]]% | |||||||

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< < | ## Tips for Series | |||||||

> > | ## Tips for Series | |||||||

- Often the hardest part of showing convergence or divergence of a series is the indecision:
*What do I believe it does?*After all, you'll have a tough time showing a series converges if it doesn't!
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< < | - The limits listed in another section can help a lot with the
*Test for Divergence.*Together with inequalities you can often get an idea of what to try to show. If the individual terms of the series "look like" as then the series "looks like" and you will want to show it diverges, perhaps even setting up a comparison, or limit comparison with 1/n itself. - Many limits boil down to "look like" ratios of polynomials after stripping out trig functions using the
*Useful Inequalities*for trig functions.
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> > | - The limits listed in UsefulLimits can help a lot with the
*Test for Divergence.*Together with inequalities you can often get an idea of what to try to show. If the individual terms of the series "look like" as then the series "looks like" and you will want to show it diverges, perhaps even setting up a comparison, or limit comparison with 1/n itself. - Many limits boil down to "look like" ratios of polynomials after stripping out trig functions using the Useful Inequalities for trig functions.
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- The eventual behavior that for any leads to the peculiar rule of thumb that in lots of ratios
*ln(n)*"looks like" 1 since any positive power of*n*will dominate it:- informally, "looks like" so converges
- more carefully, (eventually),
so, which is a convergent p-series.
-- DickFurnas - 21 Oct 2008 |

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