MATH1051 Lecture Notes - Lecture 18: Arithmetic Progression
Series
What are series?
●.. ..∑
∞
k=1
ak=a1+a2+a3+ . + an+an+1 + .
●When you add everything up, what do you get?
●This question is the series convergence check
●Add first however many finite numbers
○Partial sums
○bn= ∑
n
k=1
ak
○What does this mean?
■The nth partial sum is the sum of first n terms
●A series converges if is a number lim
n→∞ bn
Example #1
●Think of a cake
○Day 1: you eat ½ of the cake = a1
○Day 2: you eat half of the remaining half, which is ¼ of the cake = a2
○Day 3: you eat half of the remaining half, which is ⅛ of the cake = a3
○Day 4: repeat; 1/16 of the cake = a4
○Day n: =
1
2nan
●Check after finite amount of days using an
●Total cake eaten is partial sum
○/2d1= 1
○/2 /4 /4d2= 1 + 1 = 3
○/2 /4 /8 /8d3= 1 + 1 + 1 = 7
○dn= ∑
n
k=1
1
2k= 1 − 1
2n
○ lim
n→∞ 1 − 1
2n= 1 = ∑
∞
k=1
1
2k
Geometric Series formula
●If , let’s see what ran=a1n−1 r?∑
n
k=1
a1k−1 =
○r r r r ... a r a r bn= ∑
n
k=1
a1k−1 =a1+a1+a12 +a131n−2 + 1n−1
○b r r r a r ... r r ,a rr n=a1+a12+a13+ 14+a1n−1 +a1n−1 1n
●Subtracting the two, we’d end up with…
●b rbn−rn=a1−a1n
○(1 ) (1 )bn−r=a−rn
○bn=1−r
a(1−r)
1n
●r∑
∞
k=1
a1k−1 = lim
n→∞ bn= lim
n→∞ 1−r
a(1−r)
1n
○, when =a1
1−rr
| | ≤ 1
○ivergent,when = d r
| | ≥ 1
○a1
1−r
Example #2
●( )∑
∞
k=1
1
2k= ∑
∞
k=1 2
1
2
1k−1
○, /2a1= 1 1/2r=
○, so the geometric formula applies r
| | ≤ 1
○Plug in the values for the geometric series formula
○a
1−r
●1/2
1−1/2 = 1
Document Summary
What are series? k=1 ak = a1 + a2 + a3 + . This question is the series convergence check. The nth partial sum is the sum of first n terms. A series converges if lim n bn is a number. Day 1: you eat of the cake = a1. Day 2: you eat half of the remaining half, which is of the cake = a2. Day 3: you eat half of the remaining half, which is of the cake = a3. Day 4: repeat; 1/16 of the cake = a4. Check after finite amount of days using an. Total cake eaten is partial sum d1 = 1. /2 d2 = 1 + 1 = 3. 2k n k=1 a1 r k 1 = n bn = k=1 a1 r k 1 = a1 + a1 + a1 r r. 4 + a1 r n 1 + a1 r n 1 1.
