Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 100 minutes, as an unvalidated planning estimate. This collection contains 5 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Define the comparison and its domain before calculating. A finite numerical table is not a proof about all sufficiently large inputs.
State when you use a supplied theorem. Distinguish a sufficient threshold from the smallest threshold that actually works.
Attempt the changed cases before reading solutions. Written explanations need review against their rubrics.
Loading saved assessment responses…
0 of 5 tasks have recorded responses. No overall mastery score is assigned.
Define T(n)=Σ from k=1 through n of (2k²−k), with T(0)=0. Recover its degree-at-most-three polynomial from an increment identity and certify it. Give its leading asymptotic equivalent. A proposed fit adds n(n−1)(n−2)(n−3) to your polynomial: determine the first positive mismatch and explain why the four original samples do not reject it. Finally, bound Σ k^(3/2) by a positive final block and obtain its leading constant using integrals; do not claim that this fractional-power sum is the same polynomial problem.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Use the alternating increment-and-multiply construction with F0=G0=1 and multipliers r_n=2n+1 for odd n and r_n=n²+2 for even n. Prove positivity and strict increase, then classify both O directions with explicit tail-witness reasoning. Change only even multipliers to3 and classify again. Finally prove, for general finite multipliers r_n≥2, that F=O(G) holds exactly when the odd multipliers are eventually bounded, and G=O(F) exactly when the even multipliers are eventually bounded.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let H_n=Σ from j=1 to n of 1/j and T_n=Σ from k=1 to n of H_k, for integers n≥1. Count the appearances of each 1/j to prove T_n=(n+1)H_n−n. Use ln(n+1)≤H_n≤1+ln n to bound T_n on both sides and prove T_n~n ln n as n tends to infinity. The harmonic lesson proves γ=lim(H_n−ln n) exists with 0<γ<1. Using that result, find lim(T_n/n−ln n). Does the equivalent n ln n have an additive error tending to zero? Determine the order of the absolute error.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For n≥1 define P_n=∏ from k=1 to n of (1+2/k). Obtain an exact formula, an asymptotic equivalent and the limit of ln(P_n)/ln(n²) for n≥2. Does equivalence of these logarithms imply P_n~n²? Next let Q_n=(1+1/n)^n. Using x/(1+x)≤ln(1+x)≤x for x≥0 (derive it by integrating 1/(1+t) on [0,x]), prove Q_n→e. Explain why replacing each of its n factors by an asymptotically equivalent factor 1 gives the wrong limit. What happens for n² such factors?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Use the explicitly supplied remainder theorem n!=S_n exp(θ_n), where S_n=√(2πn)(n/e)^n and 0≤θ_n≤1/(12n). Bound separately the error (n!−S_n)/S_n and the error (n!−S_n)/n!. Find the smallest integer threshold certified by each bound for error at most 1%, and explain why neither is automatically the actual smallest working n. Prove n!~S_n. Finally, explain why the weaker rectangle bounds n ln n−n+1≤ln(n!)≤n ln n−n+1+ln n alone cannot establish this same relative approximation.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
Define T(n)=Σ from k=1 through n of (2k²−k), with T(0)=0. Recover its degree-at-most-three polynomial from an increment identity and certify it. Give its leading asymptotic equivalent. A proposed fit adds n(n−1)(n−2)(n−3) to your polynomial: determine the first positive mismatch and explain why the four original samples do not reject it. Finally, bound Σ k^(3/2) by a positive final block and obtain its leading constant using integrals; do not claim that this fractional-power sum is the same polynomial problem.
Use the alternating increment-and-multiply construction with F0=G0=1 and multipliers r_n=2n+1 for odd n and r_n=n²+2 for even n. Prove positivity and strict increase, then classify both O directions with explicit tail-witness reasoning. Change only even multipliers to3 and classify again. Finally prove, for general finite multipliers r_n≥2, that F=O(G) holds exactly when the odd multipliers are eventually bounded, and G=O(F) exactly when the even multipliers are eventually bounded.
Let H_n=Σ from j=1 to n of 1/j and T_n=Σ from k=1 to n of H_k, for integers n≥1. Count the appearances of each 1/j to prove T_n=(n+1)H_n−n. Use ln(n+1)≤H_n≤1+ln n to bound T_n on both sides and prove T_n~n ln n as n tends to infinity. The harmonic lesson proves γ=lim(H_n−ln n) exists with 0<γ<1. Using that result, find lim(T_n/n−ln n). Does the equivalent n ln n have an additive error tending to zero? Determine the order of the absolute error.
For n≥1 define P_n=∏ from k=1 to n of (1+2/k). Obtain an exact formula, an asymptotic equivalent and the limit of ln(P_n)/ln(n²) for n≥2. Does equivalence of these logarithms imply P_n~n²? Next let Q_n=(1+1/n)^n. Using x/(1+x)≤ln(1+x)≤x for x≥0 (derive it by integrating 1/(1+t) on [0,x]), prove Q_n→e. Explain why replacing each of its n factors by an asymptotically equivalent factor 1 gives the wrong limit. What happens for n² such factors?
Use the explicitly supplied remainder theorem n!=S_n exp(θ_n), where S_n=√(2πn)(n/e)^n and 0≤θ_n≤1/(12n). Bound separately the error (n!−S_n)/S_n and the error (n!−S_n)/n!. Find the smallest integer threshold certified by each bound for error at most 1%, and explain why neither is automatically the actual smallest working n. Prove n!~S_n. Finally, explain why the weaker rectangle bounds n ln n−n+1≤ln(n!)≤n ln n−n+1+ln n alone cannot establish this same relative approximation.
