Infinite Series
"p" Series
| The p Series | ||
| The other basic type of series is called the p Series. | ||
| It looks like this ... | ||
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"p" is just some constant. |
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| Notice that with this one we start with n = 1, not n = 0. | ||
| Notice also that we show n in a denominator. | ||
| The deal is, if n is in the numerator, the number will be bigger than 1. | ||
| Adding up a bunch of these to ANY exponent greater than 0 will go to infinity | ||
| so those are kind of no-brainers. | ||
| Notice also that since the numerator is 1, | ||
| showing the exponent on the denominator or the whole fraction | ||
| is really the same thing. | ||
| That is ... | ||
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| Some examples of this kind of series are ... | ||
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| This type of series adds up to a number if p > 1. | ||
| The p series looks a lot like a geometric series. | ||
| Ones that converge to a number look really similar. | ||
| But they are really two different things. | ||
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| The difference gets important when you calculate | ||
| the number to which the p Series totals. | ||
| The value of a p Series is totaled in a completely different way. | ||
| What we use is an integral. | ||
| So for the last p series ... | ||
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| You can change the n to an X if it makes you feel better. | ||
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| We could have used an integral on the geometric example too, | ||
| but the formula we did use is much easier. | ||
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copyright 2005 Bruce Kirkpatrick |
