Sequences identity

Question

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 $$ log\text{(}a_{n}+\text{1})\approx a_{n} $$ when $$ a_{n}\to 0 $$ , then find a sequence equivalent to $$ log_{a}\text{(}a_{n}+\text{1}) $$ when $$ a_{n}\to 0 $$

Answer 1

$$1/(1+a_n)=1-a_n+(a_n)^2+(a_n)^3-...$$ 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.

Answer 2

For the first part: As the derivative of the natural logarithm is the reciprocal, we conclude by the Mean Value Theorem that $$ \ln(1+x)-\ln 1 = x\cdot \frac1{1+\xi}$$ for some $\xi$ between $0$ and $x$.

For the second part, express $\log_a x$ in terms of $\ln$