How to prove that f(x)−f(x−1) approaches (log_(10)(10))/(log_(10)(e))?

figoveck38

figoveck38

Answered question

2022-11-03

How to prove that f ( x ) f ( x 1 ) approaches log 10 ( 10 ) log 10 ( e ) ?
Let
f ( x ) = n = 1 10 x 1 n
I noticed that as x approaches ,
f ( x ) f ( x 1 ) 2.3025. After a bit of experimenting, I found that 2.3025... = log 10 ( 10 ) log 10 ( e )
How can I prove that as x approaches , f ( x ) f ( x 1 ) approaches log 10 ( 10 ) log 10 ( e ) ?

Answer & Explanation

ontzeidena8a

ontzeidena8a

Beginner2022-11-04Added 17 answers

f ( x ) ln 10 x + γ
Subtracting, we get
f ( x ) f ( x 1 ) ln 10 x ln 10 x 1 = x ln 10 ( x 1 ) ln 10 = ln 10
Lastly, we note that
log 10 10 log 10 e = 1 ln e ln 10 = ln 10

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