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I have no idea how to solve recurrence relations of the following kind (if they can even be solved), can anyone help me? This arose in my climate change class, it isn't homework but it would greatly simplify my work if I can find a closed form for $c_n$. the relation is this $$C_n=C_{n-1}+g(n)$$ Where $g(n)$ is equal to $$g(n)=-.4+\frac{1+(\frac{.2}{n-34})}{1+(n-34)^\frac{1}{2}}$$

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    $\begingroup$ This is very very unlikely to have a closed form solution... $\endgroup$ – Lukas Geyer Sep 18 '15 at 0:02
  • $\begingroup$ Might as well ask, but I do know this $\endgroup$ – JacksonFitzsimmons Sep 18 '15 at 0:03
  • $\begingroup$ So even though we all know that this one probably isn't solvable, what are some methods for solving recurrence relations? Or should I make that a separate question? $\endgroup$ – JacksonFitzsimmons Sep 18 '15 at 7:34
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This is just:

$$ C_n - C_{n - 1} = g(n) $$

so that adding over $k = 1$ to $n$ gives:

$$ C_n = C_0 + \sum_{1 \le k \le n} g(k) $$

As stated, this has no a closed form. But using e.g. the Euler-Maclaurin formula you might get an asymptotic approximation to the value for largeish $n$.

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