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Mercator series

The Mercator series gives a taylor series for the natural logarithm

 

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Explanation

The identity for power series is

We convert this into a more useful formula and substitute r = – t

Now we divide by 1 + t and get

This is

For |t| < 1 and n → ∞ holds tⁿ⁺¹→ 0, so that

This is an elegant formula that we can easily integrate

Calculating this inverse derivative of the logarithm then gives

 

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Example 1

You can see that ln (1) = 0, because

ln (1) = 0 − 0 + 0 − 0 + ⋯ = 0

 

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History The German mathematician Nicholas Mercator used the Latin name "logarithmus naturalis" in his treatise Logarithmo-technica in 1668.

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