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  1. Use [link] , [link] , [link] , and properties of limits to prove the preceding theorem.
  2. Let f be defined and improperly-integrable on ( a , b ) . Show that, given an ϵ > 0 , there exists a δ > 0 such that for any a < a ' < a + δ and any b - δ < b ' < b we have | a a ' f | + | b ' b f | < ϵ .
  3. Let f be improperly-integrable on an open interval ( a , b ) . Show that, given an ϵ > 0 , there exists a δ > 0 such that if ( c , d ) is any open subinterval of ( a , b ) for which d - c < δ , then | c d f | < ϵ . HINT: Let { x i } be a partition of [ a , b ] such that f is defined and improperly-integrable on each subinterval ( x i - 1 , x i ) . For each i , choose a δ i using part (b). Now f is bounded by M on all the intervals [ x i - 1 + δ i , x i - δ i ] , so δ = ϵ / M should work there.
  4. Suppose f is a continuous function on a closed bounded interval [ a , b ] and is continuously differentiable on the open interval ( a , b ) . Prove that f ' is improperly-integrable on ( a , b ) , and evaluate a b f ' . HINT: Fix a point c ( a , b ) , and use the Fundamental Theorem of Calculus to show that the two limits exist.
  5. (Integration by substitution again.) Let g : [ c , d ] [ a , b ] be continuous on [ c , d ] and satisfy g ( c ) = a and g ( d ) = b . Suppose there exists a partition { x 0 < x 1 < ... < x n } of the interval [ c , d ] such that g is continuously differentiable on each subinterval ( x i - 1 , x i ) . Prove that g ' is improperly-integrable on the open interval ( c , d ) . Show also that if f is continuous on [ a , b ] , we have that
    a b f ( t ) d t = c d f ( g ( s ) ) g ' ( s ) d s .
    HINT: Integrate over the subintervals ( x i - 1 , x i ) , and use part (d).

REMARK Note that there are parts of [link] and [link] that are not asserted in [link] . The point is that these other properties do not hold forimproperly-integrable functions on open intervals. See the following exercise.

  1. Define f to be the function on [ 1 , ) given by f ( x ) = ( - 1 ) n - 1 / n if n - 1 x < n . Show that f is improperly-integrable on ( 1 , ) , but that | f | is not improperly-integrable on ( 1 , ) . (Compare this with part (4) of [link] .) HINT: Verify that 1 N f is a partial sum of a convergent infinite series, and then verify that 1 N | f | is a partial sum of a divergent infinite series.
  2. Define the function f on ( 1 , ) by f ( x ) = 1 / x . For each positive integer n , define the function f n on ( 1 , ) by f n ( x ) = 1 / x if 1 < x < n and f n ( x ) = 0 otherwise. Show that each f n is improperly-integrable on ( 1 , ) , that f is the uniform limit of the sequence { f n } , but that f is not improperly-integrable on ( 1 , ) . (Compare this with part (5) of Theorem 5.6.)
  3. Suppose f is a nonnegative real-valued function on the half-open interval ( a , ) that is integrable on every closed bounded subinterval [ a , b ' ] . For each positive integer n a , define y n = a n f ( x ) d x . Prove that f is improperly-integrable on [ a , ) if and only if the sequence { y n } is convergent. In that case, show that a f = lim y n .

We are now able to prove an important result relating integrals over infinite intervals and convergence of infinite series.

Let f be a positive function on [ 1 , ) , assume that f is integrable on every closed bounded interval [ 1 , b ] , and suppose that f is nonincreasing; i.e., if x < y then f ( x ) f ( y ) . For each positive integer i , set a i = f ( i ) , and let S N denote the N th partial sum of the infinite series a i : S N = i = 1 N a i . Then:

  1. For each N , we have
    S N - a 1 1 N f ( x ) d x S N - 1 .
  2. For each N , we have that
    S N - 1 - 1 N f ( x ) d x a 1 - a N a 1 ;
    i.e., the sequence { S N - 1 - 1 N f } is bounded above.
  3. The sequence { S N - 1 - 1 N f } is nondecreasing.
  4.  (Integral Test) The infinite series a i converges if and only if the function f is improperly-integrable on ( 1 , ) .

For each positive integer N , define a step function k N on the interval [ 1 , N ] as follows. Let P = { x 0 < x 1 < ... < x N - 1 } be the partition of [ 1 , N ] given by the points { 1 < 2 < 3 < ... < N } , i.e., x i = i + 1 . Define k N ( x ) to be the constant c i = f ( i + 1 ) on the interval [ x i - 1 , x i ) = [ i , i + 1 ) . Complete the definition of k N by setting k N ( N ) = f ( N ) . Then, because f is nonincreasing, we have that k N ( x ) f ( x ) for all x [ 1 , N ] . Also,

1 N k N = i = 1 N - 1 c i ( x i - x i - 1 ) = i = 1 N - 1 f ( i + 1 ) = i = 2 N f ( i ) = i = 2 N a i = S N - a 1 ,

which then implies that

S N - a 1 = 1 N k N ( x ) d x 1 N f ( x ) d x .

This proves half of part (1).

For each positive integer N > 1 define another step function l N , using the same partition P as above, by setting l N ( x ) = f ( i ) if i x < i + 1 for 1 i < N , and complete the definition of l N by setting l N ( N ) = f ( N ) . Again, because f is nonincreasing, we have that f ( x ) l N ( x ) for all x [ 1 , N ] . Also

1 N l N = i = 1 N - 1 f ( i ) = i = 1 N - 1 a i = S N - 1 ,

which then implies that

1 N f ( x ) d x 1 N l N ( x ) d x = S N - 1 ,

and this proves the other half of part (1).

It follows from part (1) that

S N - 1 - 1 N f ( x ) d x S N - 1 - S N + a 1 = a 1 - a N ,

and this proves part (2).

We see that the sequence { S N - 1 - 1 N f } is nondecreasing by observing that

S N - 1 N + 1 f - S N - 1 + 1 N f = a N - N N + 1 f = f ( N ) - N N + 1 f 0 ,

because f is nonincreasing.

Finally, to prove part (4), note that both of the sequences { S N } and { 1 N f } are nondecreasing. If f is improperly-integrable on [ 1 , ) , then lim N 1 N f exists, and S N a 1 + 1 f ( x ) d x for all N , which implies that a i converges by [link] . Conversely, if a i converges, then lim S N exists. Since 1 N f ( x ) d x S N - 1 , it then follows, again from [link] , that lim N 1 N f ( x ) d x exists. So, by the preceding exercise, f is improperly-integrable on [ 1 , ) .

We may now resolve a question first raised in [link] . That is, for 1 < s < 2 , is the infinite series 1 / n s convergent or divergent? We saw in that exercise that this series is convergent if s is a rational number.

  1. Let s be a real number. Use the Integral Test toprove that the infinite series 1 / n s is convergent if and only if s > 1 .
  2. Let s be a complex number s = a + b i . Prove that the infinite series 1 / n s is absolutely convergent if and only if a > 1 .

Let f be the function on [ 1 , ) defined by f ( x ) = 1 / x .

  1. Use [link] to prove that the sequence { i = 1 N 1 i - ln N } converges to a positive number γ 1 . (This number γ is called Euler's constant.) HINT: Show that this sequence is bounded above and nondecreasing.
  2. Prove that
    i = 1 ( - 1 ) i + 1 i = ln 2 .
    HINT: Write S 2 N for the 2 N th partial sum of the series. Use the fact that
    S 2 N = i = 1 2 N 1 i - 2 i = 1 N 1 2 i .
    Now add and subtract ln ( 2 N ) and use part (a).

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Source:  OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1
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