Section 11.2 Series

Review

Questions

Which of the following sequences converge? To what do they converge?

(Subscript[{((2n+5)/(3n-1))}, n=1]^∞)

(Subscript[{((6^n)/(11^n))}, n=0]^∞)

(Subscript[{(((-1))^n)((3(n^2)+n-1)/((n^2)+5))}, n=0]^∞)

(Subscript[{(((-(5/8)))^n)}, n=2]^∞)

(Subscript[{((((4/3)))^n)}, n=1]^∞)

Monotonic sequences

Definition

A sequence that is either increasing or decreasing is said to be monotonic.

Technique 3

A bounded, monotonic sequence converges.

Questions

Consider the sequence {((n+1)/n)}. Write out the first few terms of this sequence.

Is this sequence monotonic?

Is this sequence bounded?

Questions

Consider the sequence {(n/(3^n))}. Write out the first few terms of this sequence.

Is this sequence monotonic?

Is this sequence bounded?

Infinite sums

Many quantities can be written as infinite sums or series.

Example

π=3.14159265…=3+(1/10)+(4/(10^2))+(1/(10^3))+(5/(10^4))+(9/(10^5))+(2/(10^6))+(6/(10^7))+(5/(10^8))+…

Questions

What is the decimal version of (1/3)?

Write (1/3) as an infinite sum.

Write (1/3) as an infinite sum using summation notation.

As you go farther out in the sum what is happening to the individual terms?

Questions

What is the value of the infinite sum (1/2)+(1/2)+(1/2)+(1/2)+…?

Write this infinite sum using summation notation.

As you go farther out in the sum what is happening to the individual terms?

Partial sums

The partial sums for the series (Subscript[a, 1])+(Subscript[a, 2])+(Subscript[a, 3])+(Subscript[a, 4])+… are
	(Subscript[S, 1])=(Subscript[a, 1])
(Subscript[S, 2])=(Subscript[a, 1])+(Subscript[a, 2])
(Subscript[S, 3])=(Subscript[a, 1])+(Subscript[a, 2])+(Subscript[a, 3])
(Subscript[S, 4])=(Subscript[a, 1])+(Subscript[a, 2])+(Subscript[a, 3])+(Subscript[a, 4])
	⋮

Questions

What are the first 4 partial sums for the infinite series for 
	π=3+(1/10)+(4/(10^2))+(1/(10^3))+(5/(10^4))+(9/(10^5))+(2/(10^6))+(6/(10^7))+(5/(10^8))+…?

What do these partial sums represent?

Question

Write the (k^th) partial sum for (Subscript[a, 1])+(Subscript[a, 2])+(Subscript[a, 3])+(Subscript[a, 4])+… using the summation notation.

Definition

The infinite sum (Underoverscript[Σ, (k=1), ∞])(Subscript[a, k]) converges if and only if its partial sums converge. If it converges, its value is the limit of the partial sums, (Underoverscript[Σ, (k=1), ∞])(Subscript[a, k])=(lim_(n->∞))(Underoverscript[Σ, (k=1), n])(Subscript[a, k]).

Question

Consider the series (∑_(k=1)^∞)((1/k)-(1/(k+1))).

What are the first four partial sums for this series?

What is the value of the (n^th) partial sum?

Does this series converge? If so, to what value?

Geometric series

Definition

A geometric series has the form (Subscript[Σ, k=(Subscript[n, 0])]^∞)a (r^k) for some numbers (Subscript[n, 0]), a, and r.

Questions

Which of the following are geometric series?

(Subscript[Σ, k=0]^∞)2 ((((1/3)))^k)

(7/2)+(7/4)+(7/8)+(7/16)+(7/32)+(7/64)+…

(Subscript[Σ, k=1]^∞)(2^(-k))(3^k)

Questions

Consider the geometric series (Subscript[Σ, k=0]^∞)(r^k). Let (Subscript[S, n])=(Subscript[Σ, k=0]^(n-1))(r^k)=1+r+(r^2)+…+(r^(n-1)) be the (n^th) partial sum for this series.

What is r (Subscript[S, n])?

What is (Subscript[S, n])-r (Subscript[S, n])?

Solve (Subscript[S, n])-r (Subscript[S, n])=(what you got above) for (Subscript[S, n]) and determine for what values of r the partial sum converges.

Which of the geometric series converge, and for those that do converge to what value?

(Subscript[Σ, k=0]^∞)2 ((((1/3)))^k)

(7/2)+(7/4)+(7/8)+(7/16)+(7/32)+(7/64)+…

(Subscript[Σ, k=1]^∞)(2^(-k))(3^k)

Questions

Consider the geometric sum (Subscript[Σ, n=0]^∞)4((((5/6)))^n)

What are the first few terms of this sum?

What is the value of this infinite sum?

Sum[4(5/6)^n,{n,0,Infinity}]

Consider the geometric sum (Subscript[Σ, n=2]^∞)3(((-(3/5)))^n)

What are the first few terms of this sum?

Rewrite this sum in the form (Subscript[Σ, n=0]^∞)a (r^n). What is a? What is r?

What is the value of this infinite sum?

Sum[3(-3/5)^n,{n,2,Infinity}]

Consider the  sum (Subscript[Σ, n=0]^∞)(((2^n)+(5^n))/(7^n)).

What are the first few terms of this sum?

Rewrite this infinite sum as the sum of two geometric series

What is the value of this infinite sum?

Sum[(2^n+5^n)/7^n,{n,0,Infinity}]

Question

Write the repeating decimal 0.121212121212… as a geometric series and find its value as a fraction.

Divergence Theorem

Questions

Which of the following sums converge? What is true about what happens to the individual terms of the sums?

(Subscript[Σ, n=1]^∞)2=2+2+2+2+…

(Subscript[Σ, k=0]^∞) ((((1/2)))^k)=1+(1/2)+(1/4)+(1/8)+…

(Subscript[Σ, k=1]^∞)(k/(k+1))=(1/2)+(2/3)+(3/4)+(4/5)+…

Questions

Suppose the series (Subscript[a, 1])+(Subscript[a, 2])+(Subscript[a, 3])+(Subscript[a, 4])+…. converges to L.

What is (Subscript[S, n+1])-(Subscript[S, n])?

What is (lim_(n->∞))(Subscript[S, n+1])? What is (lim_(n->∞))(Subscript[S, n])? What is (lim_(n->∞))((Subscript[S, n+1])-(Subscript[S, n]))?

Theorem

If (lim_(k->∞))(Subscript[a, k])≠0, then the infinite sum (Underoverscript[Σ, k, ∞])(Subscript[a, k]) cannot converge.

Question

How do I know that (Underoverscript[Σ, (k=1), ∞])(k/(2k+1)) diverges?