# monotonic sequence, is it increasing and is it bounded? is my work right?

• September 21st 2013, 01:22 PM
abowler10154
monotonic sequence, is it increasing and is it bounded? is my work right?
an=n+1/n (btw anything that i write with a less than sign i mean less than or equal to i just cant write it.)

so i figured out that
a1=2
a2=2 1/2
a3=3 1/3

so my guess is that its increasing.

so for n=k
i need to show ak<ak+1
a1=2<ak=2 1/2 so i showed this right i believe.
next, i need to show that ak+1<ak+2
so i did (k+1)+(1/k+1)<(k+2)+(1/k+2)
cancelled out and reduced to (k+1)+1<(k+2)+1
reduced to (k+1)<(k+2)
reduced to 1<2
is this right to show that it is increasing?
then i just took the limit of n+1/n and got infinity, so it is not bounded.
is this all right? my work and all? my professor did it a different way. Thank you guys!
• September 21st 2013, 01:36 PM
emakarov
Re: monotonic sequence, is it increasing and is it bounded? is my work right?
Quote:

Originally Posted by abowler10154
so i did (k+1)+(1/k+1)<(k+2)+(1/k+2)
cancelled out and reduced to (k+1)+1<(k+2)+1

And what did you cancel out? And please don't say 1/k.

If $a_n=1+1/n$, you don't need to use the induction hypothesis, so you don't need to use induction at all. However, I am not sure the definition is not $a_n=(n+1)/n$.

In plain text, it is customary to write a_n for $a_n$ and <= for $\le$.
• September 21st 2013, 01:47 PM
abowler10154
Re: monotonic sequence, is it increasing and is it bounded? is my work right?
I made common denominators on each side, k+1 being the left common one and k+2 being the right one so i could add them individually and cancel if you understand what i mean. But is the way i did it correct also? or no.
• September 21st 2013, 01:53 PM
abowler10154
Re: monotonic sequence, is it increasing and is it bounded? is my work right?
Actually, thinking about it, i dont believe i can cancel...i think im forgetting the rules of 8th grade math in college now. So if i got to the point where i cancelled is that okay? then how would i proceed if so?
• September 21st 2013, 01:59 PM
emakarov
Re: monotonic sequence, is it increasing and is it bounded? is my work right?
It seems to me that you omitted parentheses around k+1 in 1/k+1 and treated it as a sum instead of a fraction that it is.

If $n>1$, then $n+1+\frac{1}{n+1}>n+1>n+1/n$. The last inequality holds because 1/n < 1. No use of induction hypothesis, and thus of induction, is necessary.
• September 21st 2013, 02:04 PM
Plato
Re: monotonic sequence, is it increasing and is it bounded? is my work right?
Quote:

Originally Posted by abowler10154
an=n+1/n (btw anything that i write with a less than sign i mean less than or equal to i just cant write it.)

For this kind of sequence think of the function $f(x)=x+\tfrac{1}{x}$.

Now $f'(x)=1-\tfrac{1}{x^2}$ which tells us its increasing. But looking it, it is not bounded for $x>1$.
• September 21st 2013, 02:10 PM
abowler10154
Re: monotonic sequence, is it increasing and is it bounded? is my work right?
I forgot that i can take the derivitive and if its positive its increasing. Thank you, so now how do i show the work that it is not bounded? Its obviously bounded below as a1 is 2 and its increasing. but what about being bounded above?
• September 21st 2013, 02:29 PM
Plato
Re: monotonic sequence, is it increasing and is it bounded? is my work right?
Quote:

Originally Posted by abowler10154
I forgot that i can take the derivitive and if its positive its increasing. Thank you, so now how do i show the work that it is not bounded? Its obviously bounded below as a1 is 2 and its increasing. but what about being bounded above?

Clearly it is not bounded above.
• September 21st 2013, 03:08 PM
abowler10154
Re: monotonic sequence, is it increasing and is it bounded? is my work right?