Common Mistakes

Calculus BC · AP · Infinite Sequences and Series

Focus

Radius and Interval of Convergence of Power Series

This topic is assessed on the AP Exam.

Frequent mistakes

### 1. nth-term test overreach

What it looks/sounds like: Convergence is concluded because terms → 0.

Why it happens: nth-term test is misapplied for convergence.

How to fix: Remember: nth-term can show divergence only.

### 2. Endpoint omission

What it looks/sounds like: Interval of convergence endpoints are not tested.

Why it happens: Radius result is copied without checking endpoints.

How to fix: Substitute each endpoint into the series separately.

### 3. Correct work, wrong question

What it looks/sounds like: The algebra is fine but answers a different quantity than the stem asks for.

Why it happens: Attention locks onto the first operation instead of the final ask.

How to fix: Underline the final question; after solving, verify the boxed answer matches it.

### 4. Notation and justification gaps

What it looks/sounds like: Missing units, limit language, or ‘because’ statements AP graders expect.

Why it happens: Notation is treated as decoration rather than part of the score.

How to fix: Add one justification phrase per major step (definition, theorem, or rule).

How to avoid them

1. Name the convergence test and verify conditions. 2. State Taylor center and degree explicitly. 3. Discuss error bounds when accuracy is required.

Quick self-check

• [ ] Can apply one convergence test correctly
• [ ] Can build a Taylor polynomial at a center
• [ ] Can test IOC endpoints