Calculus: Analytic Geometry and Calculus, with Vectors

12.2 Ratio test and integral test

9 Write a complete proof of the fact that the formula

x (^11199293)
x2-9

_

x

1

9

=x+x3+xs+z7+

605

is valid when IxI > 3. Obtain a similar expansion of x/(x2 + 1).
10 For each n = 1, 2, 3,. let d(n) be the number of positive integer
divisors of it, including 1 and it, so that d(1) = 1, d(2) = 2, d(3) = 2, d(4) = 3,
d(5) = 2, d(6) = 4, etcetera. Tell why the ratio test does not provide a useful
source of information about convergence of the series

d(1)x + d(2)x2 + d(3)x3 + d(4)x4 +

Tell why d(n)? 1 and the series diverges when x > 1. Tell why 1 S d(n) 5 n

and the series converges when 0 5 x < 1. Hint: The series x + 2x2 + 3x3 +

4x4 + ... is convergent when IxI < 1.

11 Give two or more examples of convergent series ul + u2 + u3 + of

positive terms for which lim does not exist. 11ns.: The series

n-*

2 + 0 2+ 22 +lO2+ 23 + 103 + 24 + 104

are simple examples.

12 Show that the series in

(1) ex -^11 =ex+e 2z +e 3x+e4 z+

is a geometric series that converges to the left member when x > 0. Use this

result to show that, when x > 0,

n

(2) X = lim I xe'kx.

ex-1 n-1 k-1

Show that if the manipulations

(3)

x n n

1

dx =

J

Jim' xekx dx = lim ( xekx dx

o ex - 1 o n-- k1 o kal

are valid, then

(4) J

x

-ldx kIIk^2

Remark: We shall soon start hearing that the last series converges to ir2/6.

13 Supposing that n is a positive integer, sketch a graph of

1

(1) + X '