Register to reply

Proving the convergence of a sequence..

by transgalactic
Tags: convergence, proving, sequence
Share this thread:
transgalactic
#1
Mar9-09, 11:55 AM
P: 1,398
[tex]2 ,2+\frac{1}{2},2+\frac{1}{2+\frac{1}{2}}[/tex]
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)

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

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
Phys.Org News Partner Science news on Phys.org
Bees able to spot which flowers offer best rewards before landing
Classic Lewis Carroll character inspires new ecological model
When cooperation counts: Researchers find sperm benefit from grouping together in mice

Register to reply

Related Discussions
Convergence of a sequence Calculus & Beyond Homework 19
Convergence of a sequence General Math 2
Proving a given sequence is a delta sequence ~ Calculus & Beyond Homework 1
Convergence of a sequence Calculus 4
Convergence of a Sequence Calculus & Beyond Homework 4