Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $s_1 = \frac{1}{4}$ and $s_{n+1} = s_n(1-s_n)$ for all $n\geq2$. Show that $\lim_{n\to\infty} ns_n =1.$

Can someone please help me to solve this problem? I have couldn't figure out.

share|improve this question

1 Answer 1

up vote 5 down vote accepted

1) $s_1>s_2>s_3>\dotsb$. let $\lim s_n =l$, then $$l=l(1-l)$$ $l=0$. that is $$\lim_{n\to\infty} s_n =0$$ so $$\lim_{n\to\infty} \frac{s_{n+1}}{s_n}=\lim_{n\to\infty}(1-s_n)=1$$ 2)Stolz's theorem

\begin{equation} \begin{split}\lim_{n\to\infty} \frac1{ns_n} &=\lim_{n\to\infty} \dfrac{\frac1{s_n}}n\\&=\lim_{n\to\infty} \left(\frac1{s_{n+1}}-\frac1{s_n}\right)\\ &=\lim_{n\to\infty} \frac{s_n-s_{n+1}}{s_{n+1}s_n}\\ &=\lim_{n\to\infty} \frac{s_n^2}{s_{n+1}s_n}\\ &=\lim_{n\to\infty} \frac{s_n}{s_{n+1}}\\&=1& \end{split} \end{equation}

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.