Unit 2 Flashcards

What is the Taylor (Maclaurin) series for e^x?

e^x = Σ_{n=0} (x^n/n!)

What is the interval of convergence for e^x?

(-∞, ∞)

What is the Taylor (Maclaurin) series for sin x?

sin x = Σ_{n=0} (-1)^n (x^{2n+1}/(2n+1)!)

What is the interval of convergence for sin x?

(-∞, ∞)

What is the Taylor (Maclaurin) series for cos x?

cos x = Σ_{n=0} (-1)^n (x^{2n}/(2n)!)

What is the interval of convergence for cos x?

(-∞, ∞)

What is the Taylor (Maclaurin) series for 1/(1 - x)?

1/(1 - x) = Σ_{n=0} (x^n)

What is the interval of convergence for 1/(1 - x)?

|x| < 1

What is the Taylor (Maclaurin) series for ln(1 + x)?

ln(1 + x) = Σ_{n=1}^{∞} (-1)^{n+1} (x^n/n)

What is the interval of convergence for ln(1 + x)?

-1 < x ≤ 1

Operations that DO NOT change the interval of convergence for a power series

1. Adding or removing finite terms
→ Behavior at infinity is unaffected

2. Differentiating term-by-term
→ Same radius; must recheck endpoints only

3. Integrating term-by-term
→ Same radius; must recheck endpoints only

4. Multiplying by a constant
→ Scalar multiplication doesn’t affect convergence

5. Reindexing the series (e.g. start at n = 1 instead of n = 0)
→ Just shifts the labeling, not the convergence

6. Adding or subtracting another power series with the same center and radius
→ Intersection of intervals is still the same

What is 6! 5!, 4!, 3!, 2!, 1! ?

6! = 720 // 5! = 120 // 4! = 24 // 3! = 6 // 2! = 2 // 1! = 1

What are the 5 types of series covered?

Power Series

Taylor Series

Alternating Series

Geometric Series

P-series

Power Series

a_k(x-c)^k

Common Test: Ratio Test

Common Questions: Interval of Convergence, Power Series Representations

Taylor Series

a_k(x-c)^k

a_k = f^k (c) / k!

Alternating Series

(-1)^n * b_k

Converges when: b_k is decreasing and lim = 0

If it doesn’t increase: inconclusive

Alternating Series Error Bound:

| s_n - S | < a_(n+1)

Geometric Series

ar^n

Defined by its common ratio

Common Test: Ratio
Common Problems: interval of convergence

p-Series

1/n^p

Converges when: p > 1

What are the 6 common tests covered

N-th term test

Direct comparison test

Alternating series test

Asymptotic comparison test

Integral comparison test

Ratio test

N-th term test

Purpose:
Quickly check if a series diverges.

Applies to:
→ Any series

Rule:

If lim n→∞ = 0, inconclusive. Otherwise, it diverges

Direct Comparison Test

Purpose:
Compare your series to a known convergent or divergent one.

Applies to:
→ Series with positive terms.

Rule:

Make sure 0 ≤ an ≤ bn
If ... (you know how it works)

Alternating Series Test

Purpose:
Check convergence for alternating series.

Applies to:
→ Series of the form

∑(−1)^n * b_n OR ∑(−1)^(n+1) * b_n

Rule:

1. bn+1≤bn eventually (terms decrease).

2. lim⁡n→∞bn=0

If BOTH are true, the series converges

Asymptotic Comparison Test

Purpose:
Compare two positive-term series when direct comparison is messy.

Applies to:
→ Series with positive terms.

C ≪lnk ≪ k^a (for 0<a<1) ≪ k^1 ≪ k^m ≪ a^k ≪ k! ≪ k^k

Rule:
If a_n ~ b_n, and b_n converges, the a_n also converges

if a_n ~ b_n, and b_n diverges, the a_n also diverges

Integral Comparison Test

Purpose:
Compare a series to an improper integral.

Applies to:
→ Series with positive, continuous, and decreasing terms f(x)f(x)f(x).

Rule:

Check image

Ratio Test

Purpose:
Used when factorials or exponentials appear.

Applies to:
→ ∑an\sum a_n∑an​ (terms can be positive or alternating).

Rule:

Check image

Typical Use:
Factorials, powers of constants, exponentials, or any time growth rates compete.