Lessons17

Question 1

What is the difference between and sequence and a series?

Answer 1

A sequence is an infinite list of numbers generated by a rule.

A series is an infinite summation of numbers. it is defined by its sequence of partial sums which is different than the original sequence.

Question 2

If the series is ½ + ¼ + ¹/₈ + ¹/₁₆ + . . . what is the rule?

Answer 2

_n=1Σ^∞ 1 /2ⁿ

Question 3

What is a sequence of Partial Sums?

Answer 3

It is the sequence of: the first term + the 1st and 2nd terms added “ the first 3 terms added, etc.

Question 4

What are the first 4 partial sums of the series _n=1Σ^∞1 / 2ⁿ? What is the formula for the individual sums?

Lesson16

Answer 4

Original Series:

½ + ¼ + ¹/₈ + ¹/₁₆ + …

Partial Sums:

½ + ¾ + ⁷/₈ + ¹⁵/₁₆

_{The formula for each sum is 1 - the last term 1 - 1 /2ⁿ:}

(1-½), (1-¼), (1-⁷/₈), (1-¹⁵/₁₆)

Question 5

What happens if the sequence of partial sums converges?

Answer 5

Then the series converges. The sequence of partial sums points to the limit.

Question 6

Prove that _n=1Σ^∞1 does not converge.

Answer 6

The sequence is 1 + 1 + 1 + . . .

The sequence os partial sums is {1, 2, 3, 4, . . .}

lim_n→∞ = ∞

Question 7

What is a telescoping series?

Answer 7

a series where all terms cancel out except for the first and last one

Question 8

Prove that _n=1Σ^∞ 1/(n(n+1) is a telescoping series.

Answer 8

Use partial fractions.

_n=1Σ^∞ [1/n - 1/(n + 1)]

_n=1Σ^∞ 1/n = 1 + ½ + ⅓ + ¼ +. . .

_n=1Σ^∞ 1/(n + 1) = ½ + ⅓ + ¼ + . . .

Subtract the two and you end up with the first and last terms: s_n = 1 - 1/(n + 1). Everything else cancels out.

Question 9

What are the 2 issues to solve for with infinite series?

Answer 9

1. Does it converge or diverge?
2. If it converges, what does it converge to?

Question 10

What is the importance of where the n starts the sequence?

Answer 10

None, really. It will affect the sums but not the divergence or convergence.

Question 11

What is the formula for a geometric series?

Answer 11

_n=0Σ^∞ arⁿ = a + ar + ar² + ar³ + . . .

a is a constant rate that can be factored out

r is the common ratio (the previous term is multiplied by r)

Question 12

What does _n=1Σ^∞ 1/n² converge to?

What does _n=1Σ^∞ 1/n³ converge to?

Answer 12

_n=1Σ^∞ 1/n² converges to π²/6,

_n=1Σ^∞ 1/n³ converges but no one knows to what

Question 13

When does a geometric series [_n=0Σ^∞arⁿ] converge? What does it converge to?

Answer 13

|r| < 1

It diverges if |r| ≥ 1

It diverges at a/(1 - r)

Question 14

Simplify and solve this geometric series:

_n=0Σ^∞ 3 /2ⁿ

Answer 14

_n=1Σ^∞3(½)ⁿ

^{a = 3, r = ½}

^{It converges at a/(1 - r) = 6}

^{Check by adding partial sums: 3 + 3/₂ + ¾ + 3/₈ + 3/16 + . . .}

Question 15

Does _n=2Σ^∞ (3/2)ⁿ diverge or converge?

Answer 15

It diverges because 3/2 > 1

Question 16

How do you calculate the sums when n does not begin at 0 or 1?

_n=3Σ^∞ (-1/5)ⁿ

Answer 16

(-1/5)³ + (-1/5)⁴ + (-1/5)⁵ + . . . .

-1/125 (1 + (-1/5) + (-1/5)² + …) Factor out (-1/5)³

a = -1/125, r = -1/5 So it converges to (-1/125)/(1 - - 1/5) = -5/750 = -1/150

Question 17

Convert the repeating decimal to a fraction. .08080808

Answer 17

(8/10)² + (8/10)⁴ + (8/10)⁶ + …

(8/10)² [1 + (1/10)² + (1/10)⁴ + …]

_n=1Σ^∞(8/10)²(1/10²)ⁿ

a = (8/10)², r = 1/10²

It converges at (8/10)² / (1 - 1/10²) = 8/100 / 99/100 = 8/99

Question 18

Is 0.99999 less than 1?

Answer 18

= 9/10 + 9/10² + 9/10³ + …

= 9/10(1 + 1/10 + 1/10² + …

a = 9/10 r = 1/10

9/10 / (1 - 1/10) = 1

A case of a real number, 1, with 2 decimal representations.

Question 19

Find the first 5 terms of the sequence of partial sums for _n=1Σ^∞ 3/[2^n-1]

Answer 19

1, 9/2, 21 /4, 45/8, 93/16