• Calculus 2
• Week1 : Sequence
  • What is a Sequence?
  • What is the Limit of a Sequence ?
    • What is an Geometric Progression?
  • What Other Properties Might a sequence Have ?
    • How Do Sequences Help with the √2 ?
    • When is a Sequence Bounded?
    • When is a Sequence Increasing?
    • What is the Monotone Convergence Theorem?
    • How Can the Monotone Convergence Theorem Help?
  • How Big Can Sequence Be ?
    • Is There a Sequence That Includes Every Integer?
    • Is There a Sequence That Includes Every Real Number between 0 and 1 ?
• Week2 : Series
  • What is a Series ? What is a Geometric Series ?
    • What does ∑a_k = L Mean ?
    • Why does ∑_k=^∞₀ (1/2)ᵏ = 2 ?
    • What is a Geometric Series?
    • What is the value of ∑_k=^∞_m rᵏ ?
  • What is a Telescoping Series ? How can I Prove That Some Series Diverge ?
    • What is the Sum of a Telescoping Series?

Calculus 2

Week1 : Sequence

What is a Sequence?

• 1,1,2,3,5,8,...
• a₁,a₂,a₃,a₄,a₅,a₆,...
• a₆=8, a₃=2
• we use (a_n) represents the whole sequence

• "arithmetic progression"
  • a sequence with a common difference between the terms.
  • 5,12,19,26,33,...
    • a_n = a₀+d_n
  • Why are these things even called arithmetic progressions?
    • Each term Is the arithmetic mean of its neighbors.
    • 12 == (5+19)/2

What is the Limit of a Sequence ?

What is an Geometric Progression?

• A geometric progression, is a sequence with a common ratio between the terms.
  • 3,6,12,24,...
• in a geometric progression, each term is the geometric mean of it's neighbors
• what is geometric mean ?
  • the geometric mean of two numbers, of a and b, is defined to be the square root = √(a·b)

What Other Properties Might a sequence Have ?

How Do Sequences Help with the √2 ?

• x₁ =1
• x_n+1 = 1/x_n + x_n/2
  • x₂ = 3/2
  • x₃ = 17/12
  • x₅ ≈ 1.41421

When is a Sequence Bounded?

• a_n is "bounded above" means there is a real number M , so that
  • for all n≥0, a_n ≤ M
• a_n is "bounded below" means there is a real number M , so that
  • for all n≥0, a_n ≥ M

When is a Sequence Increasing?

• A sequence (a_n) is increaseing if whenever m > n , then a_m > a_n
• A sequence (a_n) is decreaseing if whenever m > n , then a_m < a_n
• A sequence (a_n) is non-increaseing if whenever m > n , then a_m ≤ a_n
• A sequence (a_n) is non-decreaseing if whenever m > n , then a_m ≥ a_n
• those 4 kind of sequence are monotone

What is the Monotone Convergence Theorem?

Here's a theorem that guarantees a sequence converges.

• If the sequence (a_n) is bounded and monotone, then lim_n→∞ a_n exists.

How Can the Monotone Convergence Theorem Help?

How Big Can Sequence Be ?

Is There a Sequence That Includes Every Integer?

Yes !

• 0,-1,1,-2,2,-3,3, ...
• c_n=
  • -(n+1)/2 , if n is odd
  • n/2 , if n is even
  • starting with index 0

Is There a Sequence That Includes Every Real Number between 0 and 1 ?

No!

Week2 : Series

What is a series ? A series is basically what you get when you add up the numbers in a sequence in order.

What is a Series ? What is a Geometric Series ?

What does ∑a_k = L Mean ?

If lim_n→∞ s_n = lim_n→∞ ∑_k=ⁿ₁ a_k exists and equals L , then say

∑_k=ⁿ₁ a_k converges.

Otherwise, say ∑_k=ⁿ₁ a_k diverges.

Why does ∑_k=^∞₀ (1/2)ᵏ = 2 ?

What is a Geometric Series?

• Geometric Series : ∑_k=^∞₀ rᵏ
• let s_n = ∑_k=ⁿ₀ rᵏ
• (1-r)s_n = 1·(r⁰+r¹+...+rⁿ) - r·(r¹+r²+...+rⁿ+rⁿ⁺¹) = 1-rⁿ⁺¹
• so if r≠ 1
  • s_n = (1-r)s_n / (1-r) = (1-rⁿ⁺¹ ) / (1-r)
• so lim_n→∞ s_n = lim_n→∞ (1-rⁿ⁺¹ ) / (1-r)
• if r>1 or r<-1 , lim_n→∞ rⁿ⁺¹ is infinite
  • if -1<r<1 , lim_n→∞ rⁿ⁺¹ = 0
• 

What is the value of ∑_k=^∞_m rᵏ ?

• C·∑_k=^∞₀ rᵏ = ∑_k=^∞₀ C·rᵏ
• rᵐ·∑_k=^∞₀ rᵏ = rᵐ/(1-r) (|r|<1)
  • = ∑_k=^∞₀ r^m+k

What is a Telescoping Series ? How can I Prove That Some Series Diverge ?

What is the Sum of a Telescoping Series?

• ∑_k=^∞₁ 1/((k+1)·k)

= lim_n→∞ ∑_k=ⁿ₁ (1/k-1/(k+1))

= lim_n→∞ ( 1-1/(n+1) ) = 1