Calculus (I)
WEN-CHING LIEN
Department of Mathematics National Cheng Kung University
2008
WEN-CHINGLIEN Calculus (I)
10.2
Definition
A sequence of real numbers is a real-valued function defined on the set of positive integers
a1,a2,a3, . . . ,an, . . . {an}
WEN-CHINGLIEN Calculus (I)
10.2
Definition
A sequence of real numbers is a real-valued function defined on the set of positive integers
a1,a2,a3, . . . ,an, . . . {an}
WEN-CHINGLIEN Calculus (I)
10.2
Definition
A sequence of real numbers is a real-valued function defined on the set of positive integers
a1,a2,a3, . . . ,an, . . . {an}
WEN-CHINGLIEN Calculus (I)
(1)
1 the scalar product
{αan}={αa1, αa2, . . .} 2 the sum
{an+bn}={a1+b1,a2+b2, . . .} 3 the difference
{{an−bn}=a1−b1,a2−b2, . . .} 4 the product
{anbn}={a1b1,a2b2, . . .} 5 the reciprocal
{1/bn}={1/b1,1/b2. . .} 6 the quotient
{an/bn}={a1/b1,a2/b2, . . .}
WEN-CHINGLIEN Calculus (I)
(1)
1 the scalar product
{αan}={αa1, αa2, . . .} 2 the sum
{an+bn}={a1+b1,a2+b2, . . .} 3 the difference
{{an−bn}=a1−b1,a2−b2, . . .} 4 the product
{anbn}={a1b1,a2b2, . . .} 5 the reciprocal
{1/bn}={1/b1,1/b2. . .} 6 the quotient
{an/bn}={a1/b1,a2/b2, . . .}
WEN-CHINGLIEN Calculus (I)
(1)
1 the scalar product
{αan}={αa1, αa2, . . .} 2 the sum
{an+bn}={a1+b1,a2+b2, . . .} 3 the difference
{{an−bn}=a1−b1,a2−b2, . . .} 4 the product
{anbn}={a1b1,a2b2, . . .} 5 the reciprocal
{1/bn}={1/b1,1/b2. . .} 6 the quotient
{an/bn}={a1/b1,a2/b2, . . .}
WEN-CHINGLIEN Calculus (I)
(1)
1 the scalar product
{αan}={αa1, αa2, . . .} 2 the sum
{an+bn}={a1+b1,a2+b2, . . .} 3 the difference
{{an−bn}=a1−b1,a2−b2, . . .} 4 the product
{anbn}={a1b1,a2b2, . . .} 5 the reciprocal
{1/bn}={1/b1,1/b2. . .} 6 the quotient
{an/bn}={a1/b1,a2/b2, . . .}
WEN-CHINGLIEN Calculus (I)
(1)
1 the scalar product
{αan}={αa1, αa2, . . .} 2 the sum
{an+bn}={a1+b1,a2+b2, . . .} 3 the difference
{{an−bn}=a1−b1,a2−b2, . . .} 4 the product
{anbn}={a1b1,a2b2, . . .} 5 the reciprocal
{1/bn}={1/b1,1/b2. . .} 6 the quotient
{an/bn}={a1/b1,a2/b2, . . .}
WEN-CHINGLIEN Calculus (I)
(1)
1 the scalar product
{αan}={αa1, αa2, . . .} 2 the sum
{an+bn}={a1+b1,a2+b2, . . .} 3 the difference
{{an−bn}=a1−b1,a2−b2, . . .} 4 the product
{anbn}={a1b1,a2b2, . . .} 5 the reciprocal
{1/bn}={1/b1,1/b2. . .} 6 the quotient
{an/bn}={a1/b1,a2/b2, . . .}
WEN-CHINGLIEN Calculus (I)
(1)
1 the scalar product
{αan}={αa1, αa2, . . .} 2 the sum
{an+bn}={a1+b1,a2+b2, . . .} 3 the difference
{{an−bn}=a1−b1,a2−b2, . . .} 4 the product
{anbn}={a1b1,a2b2, . . .} 5 the reciprocal
{1/bn}={1/b1,1/b2. . .} 6 the quotient
{an/bn}={a1/b1,a2/b2, . . .}
WEN-CHINGLIEN Calculus (I)
(1)
1 the scalar product
{αan}={αa1, αa2, . . .} 2 the sum
{an+bn}={a1+b1,a2+b2, . . .} 3 the difference
{{an−bn}=a1−b1,a2−b2, . . .} 4 the product
{anbn}={a1b1,a2b2, . . .} 5 the reciprocal
{1/bn}={1/b1,1/b2. . .} 6 the quotient
{an/bn}={a1/b1,a2/b2, . . .}
WEN-CHINGLIEN Calculus (I)
(1)
1 the scalar product
{αan}={αa1, αa2, . . .} 2 the sum
{an+bn}={a1+b1,a2+b2, . . .} 3 the difference
{{an−bn}=a1−b1,a2−b2, . . .} 4 the product
{anbn}={a1b1,a2b2, . . .} 5 the reciprocal
{1/bn}={1/b1,1/b2. . .} 6 the quotient
{an/bn}={a1/b1,a2/b2, . . .}
WEN-CHINGLIEN Calculus (I)
(1)
1 the scalar product
{αan}={αa1, αa2, . . .} 2 the sum
{an+bn}={a1+b1,a2+b2, . . .} 3 the difference
{{an−bn}=a1−b1,a2−b2, . . .} 4 the product
{anbn}={a1b1,a2b2, . . .} 5 the reciprocal
{1/bn}={1/b1,1/b2. . .} 6 the quotient
{an/bn}={a1/b1,a2/b2, . . .}
WEN-CHINGLIEN Calculus (I)
(1)
1 the scalar product
{αan}={αa1, αa2, . . .} 2 the sum
{an+bn}={a1+b1,a2+b2, . . .} 3 the difference
{{an−bn}=a1−b1,a2−b2, . . .} 4 the product
{anbn}={a1b1,a2b2, . . .} 5 the reciprocal
{1/bn}={1/b1,1/b2. . .} 6 the quotient
{an/bn}={a1/b1,a2/b2, . . .}
WEN-CHINGLIEN Calculus (I)
(1)
1 the scalar product
{αan}={αa1, αa2, . . .} 2 the sum
{an+bn}={a1+b1,a2+b2, . . .} 3 the difference
{{an−bn}=a1−b1,a2−b2, . . .} 4 the product
{anbn}={a1b1,a2b2, . . .} 5 the reciprocal
{1/bn}={1/b1,1/b2. . .} 6 the quotient
{an/bn}={a1/b1,a2/b2, . . .}
WEN-CHINGLIEN Calculus (I)
Definition (monotonic sequences) The sequence {an}is said to be
–increasing if an <an+1, ∀n –nondecreasing if an≤an+1, ∀n –decreasing if an >an+1, ∀n –nonincreasing if an ≥an+1, ∀n
Example:
{−1,−2,−3, . . .} {−1,1,−1,1, . . .}
WEN-CHINGLIEN Calculus (I)
Definition (monotonic sequences) The sequence {an}is said to be
–increasing if an <an+1, ∀n –nondecreasing if an≤an+1, ∀n –decreasing if an >an+1, ∀n –nonincreasing if an ≥an+1, ∀n
Example:
{−1,−2,−3, . . .} {−1,1,−1,1, . . .}
WEN-CHINGLIEN Calculus (I)
Definition (monotonic sequences) The sequence {an}is said to be
–increasing if an <an+1, ∀n –nondecreasing if an≤an+1, ∀n –decreasing if an >an+1, ∀n –nonincreasing if an ≥an+1, ∀n
Example:
{−1,−2,−3, . . .} {−1,1,−1,1, . . .}
WEN-CHINGLIEN Calculus (I)
Definition (monotonic sequences) The sequence {an}is said to be
–increasing if an <an+1, ∀n –nondecreasing if an≤an+1, ∀n –decreasing if an >an+1, ∀n –nonincreasing if an ≥an+1, ∀n
Example:
{−1,−2,−3, . . .} {−1,1,−1,1, . . .}
WEN-CHINGLIEN Calculus (I)
Definition (monotonic sequences) The sequence {an}is said to be
–increasing if an <an+1, ∀n –nondecreasing if an≤an+1, ∀n –decreasing if an >an+1, ∀n –nonincreasing if an ≥an+1, ∀n
Example:
{−1,−2,−3, . . .} {−1,1,−1,1, . . .}
WEN-CHINGLIEN Calculus (I)
Definition (monotonic sequences) The sequence {an}is said to be
–increasing if an <an+1, ∀n –nondecreasing if an≤an+1, ∀n –decreasing if an >an+1, ∀n –nonincreasing if an ≥an+1, ∀n
Example:
{−1,−2,−3, . . .} {−1,1,−1,1, . . .}
WEN-CHINGLIEN Calculus (I)
Definition (monotonic sequences) The sequence {an}is said to be
–increasing if an <an+1, ∀n –nondecreasing if an≤an+1, ∀n –decreasing if an >an+1, ∀n –nonincreasing if an ≥an+1, ∀n
Example:
{−1,−2,−3, . . .} {−1,1,−1,1, . . .}
WEN-CHINGLIEN Calculus (I)
Definition (monotonic sequences) The sequence {an}is said to be
–increasing if an <an+1, ∀n –nondecreasing if an≤an+1, ∀n –decreasing if an >an+1, ∀n –nonincreasing if an ≥an+1, ∀n
Example:
{−1,−2,−3, . . .} {−1,1,−1,1, . . .}
WEN-CHINGLIEN Calculus (I)
Examples:
1. {an = 3n
n!}is nonincreasing?
2. {an =√
n2+1}
3. {an = n+3 ln(n+3)}
4. {an = (n+1)2 n2 } 5. {an = (2n+3n)1n}
WEN-CHINGLIEN Calculus (I)
Examples:
1. {an = 3n
n!}is nonincreasing?
2. {an =√
n2+1}
3. {an = n+3 ln(n+3)}
4. {an = (n+1)2 n2 } 5. {an = (2n+3n)1n}
WEN-CHINGLIEN Calculus (I)
Examples:
1. {an = 3n
n!}is nonincreasing?
2. {an =√
n2+1} 3. {an = n+3
ln(n+3)}
4. {an = (n+1)2 n2 } 5. {an = (2n+3n)1n}
WEN-CHINGLIEN Calculus (I)
Examples:
1. {an = 3n
n!}is nonincreasing?
2. {an =√
n2+1} 3. {an = n+3
ln(n+3)}
4. {an = (n+1)2 n2 } 5. {an = (2n+3n)1n}
WEN-CHINGLIEN Calculus (I)
Examples:
1. {an = 3n
n!}is nonincreasing?
2. {an =√
n2+1} 3. {an = n+3
ln(n+3)}
4. {an = (n+1)2 n2 } 5. {an = (2n+3n)1n}
WEN-CHINGLIEN Calculus (I)
Thank you.
WEN-CHINGLIEN Calculus (I)