Question

I'm trying to proof that c needs to be equal to 4 for the following equality.

n is all positive integers

But I'm getting 2, can anyone pin point my mistake?

\[\left| n^4-3n^6\right|<= cn^6\]
\[-n^4 + 3n^6 <= cn^6\]
\[-1+3n^2<=cn^2\]
\[-1 <= n^2(c-3)\]
\[\sqrt{-1/(c-3)} <= n\]

Answer 1

where did the negative from the absolute to n^4 come from

Answer 2

Since n represents only positive numbers then I know that, \[\left| n^4-3n^6 \right| = -n^4+3n^6\]

Answer 3

since n^4 and 3n^4 will always be positive are you trying to say?

Answer 4

The result yes and since n^6 is of a higher degree I'm setting it to be positive. For example, if n = 1 then Abs(1-3) = 2 which is the same as -1 + 3

Answer 5

Now that I look at it, I think c should be 3

Answer 6

I get 2 as well:

∣n4−3n6∣∣<=cn6, note that c >= 0
-cn6 <= n4 - 3n6 <= cn6
-cn6 + 3n6 <= n4 <= cn6 + 3n6
(3-c)n6 <= n4 <= (3+c)n6
(3-c)n2 <= 1 <= (3 +c)n2
1<= (3 + c)n2 and (3-c)n2 <= 1
1/n2 - 3 <= c and 1/n2 - 3 >= -c
c >= 1/n2 - 3 and c >= 3 - 1/n2

=> c >= 3 - 1/n2 >= 3 - 1 = 2, since smallest n = 1

Smallest possible c is 2

largest possible c is lim n -> inf 3 - 1/n2 = 3, but not quite reaching 3

so largest integer c can be is 2

Hence c = 2.

Answer 7

Though call. It clearly cannot be 2, but maybe for a large n it is but I just plugged n = 10000000000 and it wasn't even close. Using the limit I can proof c = 3. Thanks.

Answer 8

I should have taken as supremum 3 in my previous attempt. Here is another look.

∣n4−3n6∣ <= cn6, note that c >= 0 because |n4 - 3n6| >= 0, then cn6 >= 0

For n = 1 (smallest possible value)

|n4 - 3n6| = |1 - 3| = 2

But then this implies |n4 - 3n6| = 2 <= cn6 = c and c >= 2. c = 1 is not possible.

For n=2:

|n4 -3n6| = |16 - 3*64| = 16|1 - 3*4| = 16(11) < = c2^6 = 6
so c >= 16*11/64 = 11/4 ==> c >= 3, since c must be an integer

As n --> inf

0 <= |n4 - 3n6|/n6 <= c
0 <= |0 - 3| <= c
0 <= 3 <= c

Again, c >=3.

Hence least upper bound on c is 3.

Answer 9

This is what i got solving for c also, ... however i didn't change the signs because you end up with an imaginary number