Enterprise Risk Management Formula Book
2. Series expansions (for real-valued
functions)
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2.1 Exponential function and natural
logarithm (log)
function
![](I/ERMFormulaBookSeriesExpansions_files/image001.png)
![](I/ERMFormulaBookSeriesExpansions_files/image002.png)
2.2 Binomial expansion
![](I/ERMFormulaBookSeriesExpansions_files/image003.png)
where
is the binomial coefficient.
If we substitute into the binomial expansion
,
and
we
have (converges for any
if
):
![](I/ERMFormulaBookSeriesExpansions_files/image010.png)
A corollary is that:
![](I/ERMFormulaBookSeriesExpansions_files/image011.png)
2.3 Taylor series expansion
For one variable: if series converges (where
is the
’th
derivative of
and
,
etc.):
![](I/ERMFormulaBookSeriesExpansions_files/image017.png)
For more than one variable: e.g. for two variables,
if series converges (where
etc.):
![](I/ERMFormulaBookSeriesExpansions_files/image019.png)
![](I/ERMFormulaBookSeriesExpansions_files/image020.png)
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