Lessons17 Flashcards
What is the difference between and sequence and a series?
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.
If the series is ½ + ¼ + 1/8 + 1/16 + . . . what is the rule?
n=1Σ∞ 1 /2n
What is a sequence of Partial Sums?
It is the sequence of: the first term + the 1st and 2nd terms added “ the first 3 terms added, etc.
What are the first 4 partial sums of the series n=1Σ∞1 / 2n? What is the formula for the individual sums?
Lesson16
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)
What happens if the sequence of partial sums converges?
Then the series converges. The sequence of partial sums points to the limit.
Prove that n=1Σ∞1 does not converge.
The sequence is 1 + 1 + 1 + . . .
The sequence os partial sums is {1, 2, 3, 4, . . .}
limn→∞ = ∞
What is a telescoping series?
a series where all terms cancel out except for the first and last one
Prove that n=1Σ∞ 1/(n(n+1) is a telescoping series.
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: sn = 1 - 1/(n + 1). Everything else cancels out.
What are the 2 issues to solve for with infinite series?
- Does it converge or diverge?
- If it converges, what does it converge to?
What is the importance of where the n starts the sequence?
None, really. It will affect the sums but not the divergence or convergence.
What is the formula for a geometric series?
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)
What does n=1Σ∞ 1/n² converge to?
What does n=1Σ∞ 1/n3 converge to?
n=1Σ∞ 1/n² converges to π²/6,
n=1Σ∞ 1/n³ converges but no one knows to what
When does a geometric series [n=0Σ∞arn] converge? What does it converge to?
|r| < 1
It diverges if |r| ≥ 1
It diverges at a/(1 - r)
Simplify and solve this geometric series:
n=0Σ∞ 3 /2n
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 + . . .
Does n=2Σ∞ (3/2)n diverge or converge?
It diverges because 3/2 > 1
How do you calculate the sums when n does not begin at 0 or 1?
n=3Σ∞ (-1/5)n
(-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
Convert the repeating decimal to a fraction. .08080808
(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
Is 0.99999 less than 1?
= 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.
Find the first 5 terms of the sequence of partial sums for n=1Σ∞ 3/[2n-1]
1, 9/2, 21 /4, 45/8, 93/16
