Sequences identity I have some problems to find a way to prove the following statement, if someone could give me any suggestions would be grateful: Show that log(a_n+1)~~an when a_n-> 0, then find a sequence equivalent to log_a(a_n+1) when a_n-> 0

Kayla Mcdowell

Kayla Mcdowell

Answered question

2022-10-26

Sequences identity
I have some problems to find a way to prove the following statement, if someone could give me any suggestions would be grateful: Show that
l o g ( a n + 1 ) a n
when
a n 0
, then find a sequence equivalent to
l o g a ( a n + 1 )
when
a n 0

Answer & Explanation

Rene Jordan

Rene Jordan

Beginner2022-10-27Added 10 answers

1 / ( 1 + a n ) = 1 a n + ( a n ) 2 + ( a n ) 3 . . .
when a n is small enough. Now integrating we get
log ( 1 + a n ) = a n a n 2 / 2 + . . .
And owing to alternation in sign and since a n > 0 the result follows.
Sariah Mcguire

Sariah Mcguire

Beginner2022-10-28Added 2 answers

For the first part: As the derivative of the natural logarithm is the reciprocal, we conclude by the Mean Value Theorem that
ln ( 1 + x ) ln 1 = x 1 1 + ξ
for some ξ between 0 and x
For the second part, express log a x in terms of ln

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