Find lim n → ∞ ∑ p = 0 k n f ( 1 n + p )

Question

Find             lim              n        →        ∞                    ∑              p        =        0                    k        n              f          (                        1                      n            +            p                              )

Josie Stephenson · Accepted Answer

We can substract as many f(0) as we like because they are 0. IE:

lim_{n \to \infty} f (\frac{1}{n}) + f (\frac{1}{n + 1}) + \dots f (\frac{1}{n + k n}) = f (\frac{1}{n}) - f (0) + f (\frac{1}{n + 1}) - f (0) + \dots f (\frac{1}{(n + 1) k}) - f (0)

Using the formula for the derivative

f^{'} {(0)}^{+} = lim_{n \to \infty} \frac{f (0 + \frac{1}{n}) - f (0)}{\frac{1}{n}}

We can get

lim_{n \to \infty} \sum_{p = 0}^{k n} f (\frac{1}{n + p}) = lim_{n \to \infty} f^{'} (0) \frac{1}{n} \sum_{p = 1}^{n k} \frac{1}{1 + \frac{p}{n}} = f^{'} (0) lim_{n \to \infty} \frac{1}{n} (\frac{1}{1 + \frac{1}{n}} + \frac{1}{1 + \frac{2}{n}} + \dots \frac{1}{1 + k})

This Riemann sum is known to be

f^{'} (0) \int_{0}^{k} \frac{1}{1 + x} d x

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