Find the value of \(\int\limits_4^\infty  {\frac{1}{{{x^3}}}{\text{d}}x} \).

Illustrate graphically the inequality \(\sum\limits_{n = 5}^\infty  {\frac{1}{{{n^3}}}}  < \int\limits_4^\infty  {\frac{1}{{{x^3}}}{\text{d}}x}  < \sum\limits_{n = 4}^\infty  {\frac{1}{{{n^3}}}} \).

Hence write down a lower bound for \(\sum\limits_{n = 4}^\infty  {\frac{1}{{{n^3}}}} \).

Find an upper bound for \(\sum\limits_{n = 4}^\infty  {\frac{1}{{{n^3}}}} \).

\(\int\limits_4^\infty  {\frac{1}{{{x^3}}}{\text{d}}x}  = \mathop {{\text{lim}}}\limits_{R \to \infty } \int\limits_4^R {\frac{1}{{{x^3}}}{\text{d}}x} \)      (A1)

Note: The above A1 for using a limit can be awarded at any stage.
Condone the use of \(\mathop {{\text{lim}}}\limits_{x \to \infty } \).

Do not award this mark to candidates who use \(\infty \) as the upper limit throughout.

= \(\mathop {{\text{lim}}}\limits_{R \to \infty } \left[ { - \frac{1}{2}{x^{ - 2}}} \right]_4^R\left( { = \left[ { - \frac{1}{2}{x^{ - 2}}} \right]_4^\infty } \right)\)     M1

\( = \mathop {{\text{lim}}}\limits_{R \to \infty } \left( { - \frac{1}{2}\left( {{R^{ - 2}} - {4^{ - 2}}} \right)} \right)\)

\( = \frac{1}{{32}}\)     A1

[3 marks]

      A1A1A1A1

A1 for the curve
A1 for rectangles starting at \(x = 4\)
A1 for at least three upper rectangles
A1 for at least three lower rectangles

Note: Award A0A1 for two upper rectangles and two lower rectangles.

sum of areas of the lower rectangles < the area under the curve < the sum of the areas of the upper rectangles so

\(\sum\limits_{n = 5}^\infty  {\frac{1}{{{n^3}}}}  < \int\limits_4^\infty  {\frac{1}{{{x^3}}}{\text{d}}x}  < \sum\limits_{n = 4}^\infty  {\frac{1}{{{n^3}}}} \)      AG

[4 marks]

a lower bound is \(\frac{1}{{32}}\)     A1

Note: Allow FT from part (a).

[1 mark]

METHOD 1

\(\sum\limits_{n = 5}^\infty  {\frac{1}{{{n^3}}}}  < \frac{1}{{32}}\)    (M1)

\(\frac{1}{{64}} + \sum\limits_{n = 5}^\infty  {\frac{1}{{{n^3}}}}  = \frac{1}{{32}} + \frac{1}{{64}}\)     (M1)

\(\sum\limits_{n = 4}^\infty  {\frac{1}{{{n^3}}}}  < \frac{3}{{64}}\), an upper bound      A1

Note: Allow FT from part (a).

METHOD 2

changing the lower limit in the inequality in part (b) gives

\(\sum\limits_{n = 4}^\infty  {\frac{1}{{{n^3}}}}  < \int\limits_3^\infty  {\frac{1}{{{x^3}}}{\text{d}}x} \left( { < \sum\limits_{n = 3}^\infty  {\frac{1}{{{n^3}}}} } \right)\)     (A1)

\(\sum\limits_{n = 4}^\infty  {\frac{1}{{{n^3}}}}  < \mathop {{\text{lim}}}\limits_{R \to \infty } \left[ { - \frac{1}{2}{x^{ - 2}}} \right]_3^R\)     (M1)

\(\sum\limits_{n = 4}^\infty  {\frac{1}{{{n^3}}}}  < \frac{1}{{18}}\), an upper bound     A1

Note: Condone candidates who do not use a limit.

[3 marks]