Calculus II Sequence & Series: Cauchy Condensation Test by: javier
Calculus II Sequence & Series: Cauchy Condensation Test ∞
∞
∞
n=1
n=1
n=1
∑ ∑ 1∑ n n 2 a(2 ) ≤ a(n) ≤ 2n a(2n ) 2
by: javier
Understanding the Cauchy Condensation Test
The Key Idea
Understanding the Cauchy Condensation Test
The Key Idea ∞
∞
∞
n=1
n=1
n=1
∑ ∑ 1∑ n n 2 a(2 ) ≤ a(n) ≤ 2n a(2n ) 2
Understanding the Cauchy Condensation Test
The Key Idea ∞
∞
∞
n=1
n=1
n=1
∑ ∑ 1∑ n n 2 a(2 ) ≤ a(n) ≤ 2n a(2n ) 2
requires a little bit of work to prove, but well worth it..
Examples using CCT □
If a(n) is decreasing
□
and a(n) > 0 for all large n′ s
then
∑
a(n) ≈
∑
2n · a(2n )
Examples using CCT □
If a(n) is decreasing
□
and a(n) > 0 for all large n′ s
then
∑
∑ a(n) ≈ 2n · a(2n ) ∑1
example:
n
∞
∞
n=1
n=1
n=1
∑ ∑ 1∑ n n 2 a(2 ) ≤ a(n) ≤ 2n a(2n ) 2
by: javier
Understanding the Cauchy Condensation Test
The Key Idea
Understanding the Cauchy Condensation Test
The Key Idea ∞
∞
∞
n=1
n=1
n=1
∑ ∑ 1∑ n n 2 a(2 ) ≤ a(n) ≤ 2n a(2n ) 2
Understanding the Cauchy Condensation Test
The Key Idea ∞
∞
∞
n=1
n=1
n=1
∑ ∑ 1∑ n n 2 a(2 ) ≤ a(n) ≤ 2n a(2n ) 2
requires a little bit of work to prove, but well worth it..
Examples using CCT □
If a(n) is decreasing
□
and a(n) > 0 for all large n′ s
then
∑
a(n) ≈
∑
2n · a(2n )
Examples using CCT □
If a(n) is decreasing
□
and a(n) > 0 for all large n′ s
then
∑
∑ a(n) ≈ 2n · a(2n ) ∑1
example:
n
