Diver/ConvergeTestsLess18 Flashcards
What is a Geometric Series?
The sum of an infinite number of terms that have a constant ratio between successive terms.
Is n=1Σ∞ 1 /2n a geometric series?
Yes.
½ + ¼ + 1/8 + 1/16 + …
What is a sequence of partial sums?
s1 = the first term
s2 = the sum of the 1st and 2nd terms
s3 = the sum of the 1st, 2nd and 3ed terms
What can the sequence of partial sums tell you?
If the partial sums approach a fixed number, then the series converges.
The number it converges to is the Sum of the original series.
How do you calculate Partial Sums on TI nSpire?
Menu → Calculus → #5Sum
What is the Test for Convergence?
If the individual terms of the series do not get smaller, it diverges.
If they do get smaller and → 0, it converges.
If they get smaller (and you don’t know if → 0) you don’t know
When does a geometric series converge? What does it converge to?
It coverges if |r| < 1
Its sum is a/(1 - r)
Does n=1Σ∞(7/6)n converge or diverge?
It diverges because 7/6 > 1 and the terms are getting larger
Does n=1Σ∞ n/(n+1) diverge or converge?
It goes to n/n which = 1, not 0, so it diverges.
What is an example of a series where the terms go to 0, but it does not converge?
The harmonic series.
n=1Σ∞ 1/n
Does n=2Σ∞(-2/7) diverge or converge?
It converges, because |-2/7|<1
Does n=0Σ∞(cos1)n converge?
This is cos of 1 radian
cos1 ≈ .54
.54 < 1 (It’s geometric)
So, it converges
What can you do arithmetically to series?
add, multiply by a constant, subtract
Each must individually converge
n=0Σ∞(½n - 3/5n)
Both series converge and are geometric.
n=0Σ∞(½)n - n=0Σ∞(¾)n
1/(1-½) - 3/(1 - 1/5)
= 2 - 15/4 = -7/4
What is the Cantor Set?
From a line representing a set you remove one-third, then one-third each of the remaining 2 one-thirds. Keep going and you have removed all of it but have an infinite number of points left.
