Integral bounds for series

3 \includegraphics[max width=\textwidth, alt={}, center]{e313d6f0-7615-4be5-b13e-2796fd6335e5-04_540_1511_276_274} The diagram shows the curve \(\mathrm { y } = \frac { \mathrm { x } } { 2 \mathrm { x } ^ { 2 } - 1 }\) for \(x \geqslant 1\), together with a set of \(N - 1\) rectangles of unit
width. width.

1. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { N } \frac { r } { 2 r ^ { 2 } - 1 } < \frac { 1 } { 4 } \ln \left( 2 N ^ { 2 } - 1 \right) + 1$$
2. Use a similar method to find, in terms of \(N\), a lower bound for \(\sum _ { r = 1 } ^ { N } \frac { r } { 2 r ^ { 2 } - 1 }\).

6 \includegraphics[max width=\textwidth, alt={}, center]{dd0e327e-6125-4970-8cfa-cefcedfec06f-3_822_1373_264_386} The diagram shows the curve with equation \(y = \frac { 1 } { x ^ { 2 } }\) for \(x > 0\), together with a set of \(n\) rectangles of unit width, starting at \(x = 1\).

1. By considering the areas of these rectangles, explain why $$\frac { 1 } { 1 ^ { 2 } } + \frac { 1 } { 2 ^ { 2 } } + \frac { 1 } { 3 ^ { 2 } } + \ldots + \frac { 1 } { n ^ { 2 } } > \int _ { 1 } ^ { n + 1 } \frac { 1 } { x ^ { 2 } } \mathrm {~d} x$$
2. By considering the areas of another set of rectangles, explain why $$\frac { 1 } { 2 ^ { 2 } } + \frac { 1 } { 3 ^ { 2 } } + \frac { 1 } { 4 ^ { 2 } } + \ldots + \frac { 1 } { n ^ { 2 } } < \int _ { 1 } ^ { n } \frac { 1 } { x ^ { 2 } } \mathrm {~d} x$$
3. Hence show that $$1 - \frac { 1 } { n + 1 } < \sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } } < 2 - \frac { 1 } { n }$$
4. Hence give bounds between which \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } }\) lies.

7

1. By using a set of rectangles of unit width to approximate an area under the curve \(y = \frac { 1 } { x }\), show that \(\sum _ { x = 1 } ^ { \infty } \frac { 1 } { x }\) is infinite.
2. By using a set of rectangles of unit width to approximate an area under the curve \(y = \frac { 1 } { x ^ { 2 } }\), find an upper limit for the series \(\sum _ { x = 1 } ^ { \infty } \frac { 1 } { x ^ { 2 } }\).

5 \includegraphics[max width=\textwidth, alt={}, center]{e4e1c424-8dd5-4d18-9950-e902de0301b0-3_444_999_1258_539} The diagram shows the curve \(y = \frac { 1 } { x + 1 }\) together with four rectangles of unit width.

1. Explain how the diagram shows that $$\frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } + \frac { 1 } { 5 } < \int _ { 0 } ^ { 4 } \frac { 1 } { x + 1 } \mathrm {~d} x$$ The curve \(y = \frac { 1 } { x + 2 }\) passes through the top left-hand corner of each of the four rectangles shown.
2. By considering the rectangles in relation to this curve, write down a second inequality involving \(\frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } + \frac { 1 } { 5 }\) and a definite integral.
3. By considering a suitable range of integration and corresponding rectangles, show that $$\ln ( 500.5 ) < \sum _ { r = 2 } ^ { 1000 } \frac { 1 } { r } < \ln ( 1000 ) .$$

8 \includegraphics[max width=\textwidth, alt={}, center]{b9f29713-bc86-4869-9e54-195208e5e81d-5_579_1363_267_390} The diagram shows the curve with equation \(y = \frac { 1 } { x + 1 }\). A set of \(n\) rectangles of unit width is drawn, starting at \(x = 0\) and ending at \(x = n\), where \(n\) is an integer.

1. By considering the areas of these rectangles, explain why $$\frac { 1 } { 2 } + \frac { 1 } { 3 } + \ldots + \frac { 1 } { n + 1 } < \ln ( n + 1 ) .$$
2. By considering the areas of another set of rectangles, show that $$1 + \frac { 1 } { 2 } + \frac { 1 } { 3 } + \ldots + \frac { 1 } { n } > \ln ( n + 1 ) .$$
3. Hence show that $$\ln ( n + 1 ) + \frac { 1 } { n + 1 } < \sum _ { r = 1 } ^ { n + 1 } \frac { 1 } { r } < \ln ( n + 1 ) + 1$$
4. State, with a reason, whether \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r }\) is convergent.

7 \includegraphics[max width=\textwidth, alt={}, center]{63afce50-e15f-4634-b2f1-ad5d78ab8bf5-3_591_1131_986_507} The diagram shows the curve with equation \(y = \sqrt [ 3 ] { x }\), together with a set of \(n\) rectangles of unit width.

1. By considering the areas of these rectangles, explain why $$\sqrt [ 3 ] { 1 } + \sqrt [ 3 ] { 2 } + \sqrt [ 3 ] { 3 } + \ldots + \sqrt [ 3 ] { n } > \int _ { 0 } ^ { n } \sqrt [ 3 ] { x } \mathrm {~d} x$$
2. By drawing another set of rectangles and considering their areas, show that $$\sqrt [ 3 ] { 1 } + \sqrt [ 3 ] { 2 } + \sqrt [ 3 ] { 3 } + \ldots + \sqrt [ 3 ] { n } < \int _ { 1 } ^ { n + 1 } \sqrt [ 3 ] { x } \mathrm {~d} x$$
3. Hence find an approximation to \(\sum _ { n = 1 } ^ { 100 } \sqrt [ 3 ] { n }\), giving your answer correct to 2 significant figures.

7 \includegraphics[max width=\textwidth, alt={}, center]{72a1330a-c6dc-4f3a-9b0e-333b099f4509-4_782_1065_251_500} The diagram shows the curve \(y = \frac { 1 } { x }\) for \(x > 0\) and a set of \(( n - 1 )\) rectangles of unit width below the curve. These rectangles can be used to obtain an inequality of the form $$\frac { 1 } { a } + \frac { 1 } { a + 1 } + \frac { 1 } { a + 2 } + \ldots + \frac { 1 } { b } < \int _ { 1 } ^ { n } \frac { 1 } { x } \mathrm {~d} x$$ Another set of rectangles can be used similarly to obtain $$\int _ { 1 } ^ { n } \frac { 1 } { x } \mathrm {~d} x < \frac { 1 } { c } + \frac { 1 } { c + 1 } + \frac { 1 } { c + 2 } + \ldots + \frac { 1 } { d }$$

1. Write down the values of the constants \(a\) and \(c\), and express \(b\) and \(d\) in terms of \(n\). The function f is defined by \(\mathrm { f } ( n ) = 1 + \frac { 1 } { 2 } + \frac { 1 } { 3 } + \ldots + \frac { 1 } { n } - \ln n\), for positive integers \(n\).
2. Use your answers to part (i) to obtain upper and lower bounds for \(\mathrm { f } ( n )\).
3. By using the first 2 terms of the Maclaurin series for \(\ln ( 1 + x )\) show that, for large \(n\), $$f ( n + 1 ) - f ( n ) \approx - \frac { n - 1 } { 2 n ^ { 2 } ( n + 1 ) } .$$

3 The diagram shows the curve \(y = \frac { 1 } { x ^ { 3 } }\) for \(1 \leqslant x \leqslant n\) where \(n\) is an integer. A set of ( \(n - 1\) ) rectangles of unit width is drawn under the curve. \includegraphics[max width=\textwidth, alt={}, center]{736932f1-4007-4a04-a08b-2551db0b136c-2_611_947_1103_557}

1. Write down the sum of the areas of the rectangles.
2. Hence show that \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 3 } } < \frac { 3 } { 2 }\).