AP Calculus Review: Series, Taylor/Maclaurin
Polynomials, Error Bounds, and Convergence
Series of Constants
1. Geometric Series
∞
X
arn
n=1
• Diverges if r ≥ 1
• Converges if |r| < 1 with sum a
1−r
• This is the only series where you can actually find the sum.
2. P-Series
∞
X 1
p
, p>0
n=1
n
• Diverges if 0 < p ≤ 1
• Converges if p > 1
3. Alternating Series
∞
X
(−1)n an
n=1
• Diverges if limn→∞ an ̸= 0 (Nth Term Test for Divergence)
1
Polynomials, Error Bounds, and Convergence
Series of Constants
1. Geometric Series
∞
X
arn
n=1
• Diverges if r ≥ 1
• Converges if |r| < 1 with sum a
1−r
• This is the only series where you can actually find the sum.
2. P-Series
∞
X 1
p
, p>0
n=1
n
• Diverges if 0 < p ≤ 1
• Converges if p > 1
3. Alternating Series
∞
X
(−1)n an
n=1
• Diverges if limn→∞ an ̸= 0 (Nth Term Test for Divergence)
1
