transgalactic
Mar9-09, 11:55 AM
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
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