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0<x<1

Prove that lim x^n = 0, as n approaches infinity.

Prove that lim x^n = 0, as n approaches infinity.

Put d = 1/x - 1.

Use (1+d)^n >= 1 + n*d.

page 127 in palace

Use (1+d)^n >= 1 + n*d.

page 127 in palace

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

Suppose that lim_sup a(n)^(1/n) < 1.

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

Suppose that lim_sup a(n)^(1/n) < 1.

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

It converges.

(this is Cauchy's test)

(this is Cauchy's test)

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

Suppose that lim_inf a(n)^(1/n) > 1.

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

Suppose that lim_inf a(n)^(1/n) > 1.

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

It diverges.

page 130 in palace

page 130 in palace

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

Suppose that lim_sup a(n)^(1/n) = 1.

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

Suppose that lim_sup a(n)^(1/n) = 1.

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

Nothing as far as convergence goes. It can converge or diverge,
the stipulated condition is inconclusive.

series 1/n diverges

series 1/n^2 converges

series 1/n diverges

series 1/n^2 converges

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

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

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

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

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

It converges.

This is the ratio test, or D'Alembert's test.

This is the ratio test, or D'Alembert's test.

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

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

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

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

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

It diverges.

This is the ratio test, or D'Alembert's test.

This is the ratio test, or D'Alembert's test.

State and prove Cauchy's Integral Test for convergence of series.

f : [1,oo) -> [0,oo) decreasing

(1) Riemann_integral_from_1_to_x_of_f converges, as x approaches infinity

(2) series f(n) converges

thesis: (1) <=> (2)

page 133 in palace

(1) Riemann_integral_from_1_to_x_of_f converges, as x approaches infinity

(2) series f(n) converges

thesis: (1) <=> (2)

page 133 in palace

Using Cauchy's Integral Test show that the series 1/n diverges.

hint: log(x) tends to infinity, as x approaches infinity

page 133 in golden gate

page 133 in golden gate

Using Cauchy's Integral Test investigate the convergence of the
series 1/n^A, where A is a real number.

converges for A>1

diverges for A<=1

page 135 in palace

diverges for A<=1

page 135 in palace

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

(1) series b(n) converges

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

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

(1) series b(n) converges

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

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

converges