Power Series

Question 1

nth partial sum of a geometric series

Answer 1

S_n = t₁ • (1–rⁿ) / (1–r)

Question 2

A geometric series converges if |r| < 1

The number it converges to is…

Answer 2

S = t₁ • 1 / (1–r)

Question 3

If ƒ is a differentiable function, then you can write ƒ(x) as a Taylor Series expansion about x = a as…

If a = 0, the series is called a Maclaurin series

Answer 3

ƒ(x) = ƒ(a) + ƒ’(a)(x–a) + ^ƒ’‘(a)/_2! (x–a)² + ƒ’’‘(a)/3! (x–a)³ + ··· + ^ƒ(n)(a)/_n! (x–a)ⁿ + ···

Question 4

Power series of

e^x

Answer 4

∑^{<span>∞</span>}_n=0 1/n! x^{<span>n</span>}

1 + x + x²/2! + x³/3! + ···

Question 5

Power series of

sinx

Answer 5

∑^{<span>∞</span>}_n=0 (–1)ⁿ 1/(2n + 1)! x^{<span>2n+1</span>}

x – x³/3! + x⁵/5! – x⁷/7! + ···

Question 6

Power series of

cosx

Answer 6

∑^∞_n=0 (–1)ⁿ 1/(2n)! x²ⁿ

1 – x²/2! + x⁴/4! – x⁶/6! + ···

Question 7

Power series of

lnx

Answer 7

∑^∞_n=1 (–1)ⁿ⁺¹ 1/n (x – 1)ⁿ

(x–1) – 1/2(x–1)² + 1/3(x–1)³ – 1/4(x–1)⁴ + ···

Question 8

Power series of

1 / 1–x

(A geometric series)

Answer 8

∑^∞_n=0 xⁿ

1 + x + x² + x³ + x⁴ + ···

Question 9

Power series of

tan^-1x

Answer 9

∑^∞_n=0 (–1)ⁿ 1/(2n + 1) x²ⁿ⁺¹

x – x³/3 + x⁵/5 – x⁷/7 + ···

Question 10

Absolute convergence

Answer 10

The series ∑^∞_n=1 t_n converges absolutely if ∑∞n=1 |t_n| converges.

Question 11

nth term test

Answer 11

If lim_n→∞ t_n ≠ 0, then the series diverges

Question 12

Alternating series test

a series whose terms are alternately positive and negative.

Answer 12

The series ∑^∞_n=1 (–1)ⁿ⁺¹ b_n

converges if all three are satified:

1. b > 0
2. b_n > b_n+1 for all n
3. b_n → 0 as n → ∞

Question 13

Ratio Test

Answer 13

For the series ∑^{<span>∞</span>}_{<span>n=0</span>} tn,

if L = lim_{<span>n→∞</span>} | (tn+1)/(tn) |,

then the series:

converges if absolutely if L < 1

diverges if L > 1

may either converge or diverge if L = 1

Question 14

Integral test

If ƒ is positive, conintous, and decreasing for x ≥ 1, and a_n = ƒ(n), then…

Answer 14

∑^∞_n=1 a_n and ∫₁^∞ ƒ(x) dx

Either both converge or diverge

Question 15

Direct comparison test

Le 0 ≤ a_n ≤ b_n for all n.

Answer 15

If ∑^∞_n=1 b_n converges, then ∑^∞_n=1 a_n converges.

If ∑^∞_n=1 a_n diverges, then ∑^∞_n=1 b_n diverges.

Question 16

Limit comparison test

L = lim_n→∞ [ƒ(n) / g(n)]

Answer 16

• If L is a postive real number, both series either converge or diverge.
• If L = 0, and ∑g(n) converges, then ∑ƒ(n) converges.
• If L = ∞, and ∑g(n) diverges, then ∑ƒ(n) diverges.

Question 17

p-Series

Answer 17

The p-series ∑^∞_n=1 1 / n^p

converges in p > 1 and diverges if p ≤ 1.

Question 18

Remainder of a Taylor Series

If M is the maximum value of |ƒ⁽ⁿ⁺¹⁾(x)| on the interval a and x, then the Langrange error bound is

Answer 18

|R_n| ≤ M/_{( n + 1 )!} |x – a|ⁿ⁺¹