BC Series: Flowchart Strategy (Ask These Questions)
Don’t start computing. Start choosing. Follow the questions in order. Each step includes what to do if YES and what it means if NO.
Decision Flow (YES / NO)
1
Is \( \lim_{n\to\infty} a_n \neq 0 \) (or does not exist)? Term Test
YES → Diverges. Stop. (Full credit: “Since \( \lim a_n \neq 0 \), the series diverges.”)
NO → The limit is 0 → Inconclusive. You MUST continue. (Limit 0 does not mean convergent.)
2
Do the signs alternate ( \( (-1)^n \), \( (-1)^{n-1} \), etc.)? AST
YES → Write \(a_n = (-1)^n b_n\). Check: (i) \(b_n\) decreasing, (ii) \(b_n\to 0\). If both true → Converges (AST).
NO / AST fails → Don’t force AST. Move to the next question (Ratio/Comparison/Integral).
Quality move → Also check \( \sum |a_n| \): if it converges → absolute; if it diverges but AST works → conditional.
3
Do you see factorials or exponentials like \(a^n\) (or products)? Ratio Test
YES → Ratio is the gold standard: \(L=\lim\left|\frac{a_{n+1}}{a_n}\right|\).
If \(L<1\) → converges; if \(L>1\) → diverges; if \(L=1\) → inconclusive (try another test).
NO → Next question.
4
Is it basically a rational/polynomial function in \(n\)? Limit Comparison
YES → Compare “boss terms” (highest degree top & bottom). Reduce to \( \frac{1}{n^p} \) behavior.
NO → Consider Integral Test candidates (log-heavy) or other comparisons.
5
Is it log-heavy like \( \frac{1}{n(\ln n)^p} \) or \( \frac{\ln n}{n^p} \)? Integral / Comparison
YES → Integral Test is allowed only if \(f\) is positive, continuous, decreasing and \(f(n)=a_n\). Otherwise use comparison.
NO → Use an appropriate comparison (dominant growth, squeezing, etc.).
Power Series Reminder: Radius is not the interval. Find \(R\) (Ratio/Root) → then test both endpoints separately → endpoints decide brackets.
Core Cheat Sheet (with IoC + Brackets)
-
\(e^x,\ \sin x,\ \cos x\)
Always convergent
\(R=\infty,\ \text{IoC }(-\infty,\infty)\) - \(\dfrac{1}{1-x}=\sum_{n=0}^{\infty}x^n\) |x|<1
- \(\dfrac{1}{1+x}=\sum_{n=0}^{\infty}(-1)^n x^n\) |x|<1
- \(\ln(1+x)=\sum_{n=1}^{\infty}(-1)^{n-1}\dfrac{x^n}{n}\) |x|<1
Endpoint Bracket Logic:
Diverges at endpoint → use ( )
Converges at endpoint → use [ ]
After finding \(R\), ALWAYS test endpoints. Brackets come from endpoint behavior.
Diverges at endpoint → use ( )
Converges at endpoint → use [ ]
After finding \(R\), ALWAYS test endpoints. Brackets come from endpoint behavior.
Alternating Error Bound:
\( |R_N|\le b_{N+1} \)
In human language: “The error is smaller than the first term you ignore.”
\( |R_N|\le b_{N+1} \)
In human language: “The error is smaller than the first term you ignore.”
Progress Tracker
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What makes this “exam-proof”
This sheet forces the correct workflow: Reflex → Recognition → Correct Test → Justification → Endpoint & Error Mastery. Try first, then reveal the full logic (in green).
BC Series — MCQ Mastery Test (20 Questions)
Interactive practice with professional LaTeX rendering. Target 18+/20.