Proving the convergence of a sequence

1. Mar 9, 2009

transgalactic

$$2 ,2+\frac{1}{2},2+\frac{1}{2+\frac{1}{2}}$$
etc..
(the sequence consists only from positive number so the sum is not negative)
in order to prove that its convergent i need to prove monotonicity and boundedness

monotonicity:(by induction)

$$a_1=2$$
$$a_2=2.5$$
so i guess its increasing
suppose n=k is true:
$$a_{k-1}<a_k$$
prove n=k+1 ($$a_{k}<a_{k+1}$$)
$$a_k>a_{k-1}\\$$
$$\frac{1}{a_k}<\frac{1}{a_{k-1}}\\$$
$$2+\frac{1}{a_k}<2+\frac{1}{a_{k-1}}\\$$
$$a_{k+1}<a_k$$

i proved the opposite :)
so this is weird.

the answer in the book tells me to split the sequence into odd /even sub sequences
the one is ascending and the other its descending.

i cant see how many sub sequences i need to split it to
maybe its 5 or 10
what is the general way of solving it.
and how you explained that i proved the opposite

Last edited: Mar 9, 2009