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!
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.
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?
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 , then . The last inequality holds because 1/n < 1. No use of induction hypothesis, and thus of induction, is necessary.
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?