Is there a more elementary way to arrive at: ln &#x2061;<!-- ⁡ --> ( m ) = <munder>

tripes3h

tripes3h

Answered question

2022-06-30

Is there a more elementary way to arrive at: ln ( m ) = lim n k = n + 1 m n 1 k ?

Answer & Explanation

engaliar0l

engaliar0l

Beginner2022-07-01Added 13 answers

k = n + 1 m n 1 k
is a Riemann Sum of the function f ( x ) = 1 x in the interval [ 1 , m ], so
lim n k = n + 1 m n 1 k = 1 m 1 x d x = log ( m )
ttyme411gl

ttyme411gl

Beginner2022-07-02Added 6 answers

S := k = n + 1 n m 1 k = n n m d t t
so that
n n m d t t + 1 < S < n n m d t t .
The exact bounds are log ( n m + 1 n + 1 ) and log ( m )

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