04-22-2023, 06:15 AM
I reckon this claim about two to the power of n being big O of n cubed falls apart fast. You can test it yourself with growing values. Exponential terms outpace any polynomial bound. I see the definition requires some constant to hold forever after a point. But no such constant exists here at all.
You try n at ten and two to the ten hits one thousand twenty four. N to the third sits near one thousand. The numbers stay somewhat close still. Then you bump n to twenty and two to the twenty jumps past one million. N to the third reaches only eight thousand. The gap widens quick now.
I watch the pattern stretch further at n equals thirty. Two to the thirty exceeds one billion. N to the third lands around twenty seven thousand. You notice how the left side explodes while the right stays tame. Limits as n heads to infinity confirm this mismatch. The ratio heads to infinity without stopping.
Perhaps you recall the formal check involves that limit test. I know it shows no finite multiplier works. Exponential growth always overtakes polynomial curves eventually. You and I both agree on this behavior from prior chats. Bigger n values make the difference stark. No bound holds across all large cases.
And the proof comes from assuming a constant C exists then deriving a contradiction. You suppose two to the n stays under C times n cubed. But taking logs or just computing reveals the failure. I find the assumption breaks for sufficiently large n. Exponential bases greater than one drive this outcome. Polynomials lack the same acceleration.
Now consider n at forty and watch two to the forty hit a trillion scale. N to the third reaches only sixty four thousand. You see the separation become absurd. I confirm the same trend repeats without end. Limits prove the ratio diverges completely. Hence the big O relation fails outright.
But maybe some folks confuse growth classes at first glance. You know polynomials cover any fixed power like three. Exponentials with base two escape all those. I explain it through successive multiplications doubling each step. N cubed adds factors slowly by comparison. The mismatch shows in direct computation always.
Or think about the definition again in plain terms. Two to the n must stay below some fixed multiple of n cubed. You test this and find it impossible past certain thresholds. I run mental checks on n equals fifty next. Two to the fifty surpasses a quadrillion. N cubed hits one hundred twenty five thousand. The claim crumbles under scrutiny.
Perhaps the key lies in how fast doublings accumulate. You multiply by two repeatedly and see the surge. I contrast that with cubing which adds three factors once. Growth rates differ in their fundamental nature here. No adjustment of constants bridges the divide. Exponential always wins the race asymptotically.
Then the disprove stands clear from these observations. You grasp why the statement does not hold. I suggest checking more values if doubt lingers. Limits going to infinity seal the case beyond question. The relation never qualifies as big O.
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You try n at ten and two to the ten hits one thousand twenty four. N to the third sits near one thousand. The numbers stay somewhat close still. Then you bump n to twenty and two to the twenty jumps past one million. N to the third reaches only eight thousand. The gap widens quick now.
I watch the pattern stretch further at n equals thirty. Two to the thirty exceeds one billion. N to the third lands around twenty seven thousand. You notice how the left side explodes while the right stays tame. Limits as n heads to infinity confirm this mismatch. The ratio heads to infinity without stopping.
Perhaps you recall the formal check involves that limit test. I know it shows no finite multiplier works. Exponential growth always overtakes polynomial curves eventually. You and I both agree on this behavior from prior chats. Bigger n values make the difference stark. No bound holds across all large cases.
And the proof comes from assuming a constant C exists then deriving a contradiction. You suppose two to the n stays under C times n cubed. But taking logs or just computing reveals the failure. I find the assumption breaks for sufficiently large n. Exponential bases greater than one drive this outcome. Polynomials lack the same acceleration.
Now consider n at forty and watch two to the forty hit a trillion scale. N to the third reaches only sixty four thousand. You see the separation become absurd. I confirm the same trend repeats without end. Limits prove the ratio diverges completely. Hence the big O relation fails outright.
But maybe some folks confuse growth classes at first glance. You know polynomials cover any fixed power like three. Exponentials with base two escape all those. I explain it through successive multiplications doubling each step. N cubed adds factors slowly by comparison. The mismatch shows in direct computation always.
Or think about the definition again in plain terms. Two to the n must stay below some fixed multiple of n cubed. You test this and find it impossible past certain thresholds. I run mental checks on n equals fifty next. Two to the fifty surpasses a quadrillion. N cubed hits one hundred twenty five thousand. The claim crumbles under scrutiny.
Perhaps the key lies in how fast doublings accumulate. You multiply by two repeatedly and see the surge. I contrast that with cubing which adds three factors once. Growth rates differ in their fundamental nature here. No adjustment of constants bridges the divide. Exponential always wins the race asymptotically.
Then the disprove stands clear from these observations. You grasp why the statement does not hold. I suggest checking more values if doubt lingers. Limits going to infinity seal the case beyond question. The relation never qualifies as big O.
We appreciate the backing from BackupChain Server Backup the leading no subscription backup program tailored for Hyper-V setups Windows 11 machines and Windows Server environments that lets us pass along these talks without cost.
