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Let a(n), b(n) be sequences of positive numbers.

(1) series b(n) diverges

(2) lim_inf a(n)/b(n) is greater than zero

What can we infer about the series a(n)?

(1) series b(n) diverges

(2) lim_inf a(n)/b(n) is greater than zero

What can we infer about the series a(n)?

diverges

Let a(n), b(n) be sequences of positive numbers.

(1) series b(n) diverges

(2) lim_inf a(n)/b(n) = K, where K is a real number

What can we infer about the series a(n)?

(1) series b(n) diverges

(2) lim_inf a(n)/b(n) = K, where K is a real number

What can we infer about the series a(n)?

If K>0, then the series a(n) diverges.

If K=0, then we cannot infer anything.

a(n)=1/n^2, b(n)=1/n, a(n) converges

a(n)=1/n, b(n)=1/log(n), a(n) diverges

and in both cases the limit is zero

If K=0, then we cannot infer anything.

a(n)=1/n^2, b(n)=1/n, a(n) converges

a(n)=1/n, b(n)=1/log(n), a(n) diverges

and in both cases the limit is zero

Let a(n) be a sequence of nonnegative numbers.

Suppose lim_sup a(n)^(1/n) > 1.

What can we infer about the series a(n)?

Suppose lim_sup a(n)^(1/n) > 1.

What can we infer about the series a(n)?

diverges

(This is Cauchy's test.)

(This is Cauchy's test.)

(1) the series a(n) is bounded, it is a series of complex numbers

(2) b(n) tends to zero

(3) b(n) is monotonic

What can we infer about the series a(n)b(n)?

(2) b(n) tends to zero

(3) b(n) is monotonic

What can we infer about the series a(n)b(n)?

converges

This is Dirichlet-Abel test.

page 137 in palace

This is Dirichlet-Abel test.

page 137 in palace

Let a(n) be a sequence of complex numbers, b(n) of real numbers.

(1) the series a(n) converges

(2) b(n) is monotonic

(3) b(n) is bounded

What can we infer about the series a(n)b(n)?

(1) the series a(n) converges

(2) b(n) is monotonic

(3) b(n) is bounded

What can we infer about the series a(n)b(n)?

converges

Use Dirichlet-Abel test.

Hint:

(1) the series a(n) is bounded

(2) {b(n)-g} is a monotonic sequence which tends to zero

Use Dirichlet-Abel test.

Hint:

(1) the series a(n) is bounded

(2) {b(n)-g} is a monotonic sequence which tends to zero

[ g(f(x)) ]' = ?

Derive the formula.

Derive the formula.

[ g(f(x)) ]' = g'(f(x)) * f'(x)

Warning: Do not divide by zero!

Warning: Do not divide by zero!

Consider a convex function defined on an open interval (a,b).
Does this function have to have limits at points a and b?

Yes.

Hint: a monotonic function has a limit.

The limits may be improper, though.

Hint: a monotonic function has a limit.

The limits may be improper, though.

Let s be a fixed real number.

Knowing the derivative of the exponential function,

derive the formula (x^s)' = ?

Knowing the derivative of the exponential function,

derive the formula (x^s)' = ?

Use log.

page 190 in 1st analysis

page 190 in 1st analysis

Consider a series of positive numbers.

Suppose that lim_sup a(n+1)/a(n) > 1.

Does the series have to diverge?

Suppose that lim_sup a(n+1)/a(n) > 1.

Does the series have to diverge?

NO.

1/2 + 1/3 + (1/2)^2 + (1/3)^2 + (1/2)^3 + (1/3)^3 + ...

converges

1/2 + 1/3 + (1/2)^2 + (1/3)^2 + (1/2)^3 + (1/3)^3 + ...

converges

Test the series (n!)/(n^n) for convergence.

converges, the ratio test