Friday, 12 September 2014

Find the limit limntoinftyfrac1a1a2+frac1a2a3+cdots+frac1anan1




Given a1=1 and an=an1+4 where n2 calculate,
lim





First I calculated few terms a_1=1, a_2=5, a_3=9,a_4=13 etc. So
\lim_{n\to \infty }\frac{1}{a_1a_2}+\frac{1}{a_2a_3}+\cdots+\frac{1}{a_na_{n-1}}=\lim_{n\to \infty }\frac{1}{5}+\frac{1}{5\times9}+\cdots+\frac{1}{a_na_{n-1}}



Now I got stuck. How to proceed further? Should I calculate the sum ? Please help.


Answer



HINT:



\dfrac4{a_ma_{m-1}}=\dfrac{a_m-a_{m-1}}{a_ma_{m-1}}=?




a_m=1+4\cdot(m-1)=?



Do you recognize the Telescoping series?


No comments:

Post a Comment

real analysis - How to find lim_{hrightarrow 0}frac{sin(ha)}{h}

How to find \lim_{h\rightarrow 0}\frac{\sin(ha)}{h} without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...