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content/blog/haskellrealsequences.md

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+---
+title: "Real Analysis Sequences in Haskell"
+date: 2019-05-21T22:18:21-04:00
+draft: false
+---
+
+In Real Analysis it is useful to look at terms of a sequence. One of the best ways I've found to do this is in believe it or not Haskell. This is mainly for these two reasons
+
+- Support for infinite data structures
+
+- Built-in Data Type to keep fractional precision
+
+## Code
+
+Let's get started, first let us define a sequence by the following:
+$$
+f(1) = 1, f(2) = 2, f(n) = \frac{1}{2}(f(n - 2) + f(n - 1))
+$$
+That is equivalent to the following haskell code:
+
+```haskell
+f :: Integral a => a -> Ratio a
+f 1 = 1
+f 2 = 2
+f n = 0.5 * (f (n - 2) + f (n - 1))
+```
+
+Now to generate the sequence we just need to map $f$ onto the natural numbers.
+
+```haskell
+nsequence = map f [1..]
+```
+
+If you want to look at specific subsequences, such as even or odd:
+
+```haskell
+odd_generator n = 2 * n - 1
+odds = map odd_generator [1..]
+
+even_generator n = 2 * n
+evens = map odd_generator [1..]
+```
+
+To look at the differences between each term:
+
+```haskell
+diff x = map (\(a, b) -> a - b) $ zip (tail x) (init x)
+```
+