Your sequence is ,
It is not monotone but it is bounded.
Hint: look at the odd subscripted terms and then the even.
(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
so i guess its increasing
suppose n=k is true:
prove n=k+1 ( )
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
i understood that they are not increasing nor decreasing.
so i will just have to check the first few members instead
of the first two.
my remaining question is ,how did you decide to split it into two
sub sequences (by odd index and even index)
there are infinite number of sub sequences
why just these two?