I calculated and guessed well . In fact I noticed the coefficient in front of can be written as a sequence that tends to 1 when n tends to positive infinite.

I found that . I set . I'm completely despaired. I need help on this problem. How can I prove that the sequence tends to ? The first terms are , , , , etc. I know it's a famous sequence, where each step is half closer to 1 than the precedent.

But I need to be rigorous, I can't say " I'm seeing that the coefficients of the method are like this sequence". I must prove it and I'm not able.

Is there any other way I'm not seeing to do the first question? I'm lost.