| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Integral bounds for series |
| Difficulty | Challenging +1.2 This is a structured Further Maths question on integral bounds for series with clear visual guidance. While it requires understanding the relationship between Riemann sums and integrals, each part is carefully scaffolded. The integration is routine (∫1/x² dx), and the algebraic manipulation to obtain bounds is straightforward. More challenging than a standard C3 question due to the proof element and Further Maths context, but the step-by-step structure makes it accessible. |
| Spec | 1.08g Integration as limit of sum: Riemann sums |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Area under graph \((\int 1/x^2 dx\), 1 to \(n+1\)) < Sum of rectangles (from 1 to \(n\)) | B1 | Sum (total) seen or implied eg diagram; accept areas (of rectangles) |
| Area of each rectangle = Width × Height = \(1 \times 1/x^2\) | B1 | Some evidence of area worked out – seen or implied |
| (ii) Indication of new set of rectangles | B1 | |
| Similarly, area under graph from 1 to \(n\) > sum of areas of rectangles from 2 to \(n\) | B1 | Sum (total) seen or implied |
| Clear explanation of A.G. | B1 | Diagram; use of left-shift of previous areas |
| (iii) Show complete integrations of RHS, using correct, different limits | M1 | Reasonable attempt at \(\int x^{-2} dx\) |
| Correct answer, using limits, to one integral | A1 | |
| Add 1 to their second integral to get complete series | M1 | |
| Clearly arrive at A.G. | A1 | |
| (iv) Get one limit | B1 | Quotable |
| Get both 1 and 2 | B1 | Quotable; limits only required |
**(i)** Area under graph $(\int 1/x^2 dx$, 1 to $n+1$) < Sum of rectangles (from 1 to $n$) | B1 | Sum (total) seen or implied eg diagram; accept areas (of rectangles)
Area of each rectangle = Width × Height = $1 \times 1/x^2$ | B1 | Some evidence of area worked out – seen or implied
**(ii)** Indication of new set of rectangles | B1 |
Similarly, area under graph from 1 to $n$ > sum of areas of rectangles from 2 to $n$ | B1 | Sum (total) seen or implied
Clear explanation of A.G. | B1 | Diagram; use of left-shift of previous areas
**(iii)** Show complete integrations of RHS, using correct, different limits | M1 | Reasonable attempt at $\int x^{-2} dx$
Correct answer, using limits, to one integral | A1 |
Add 1 to their second integral to get complete series | M1 |
Clearly arrive at A.G. | A1 |
**(iv)** Get one limit | B1 | Quotable
Get both 1 and 2 | B1 | Quotable; limits only required6\\
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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 $x = 1$.\\
(i) By considering the areas of these rectangles, explain why
$$\frac { 1 } { 1 ^ { 2 } } + \frac { 1 } { 2 ^ { 2 } } + \frac { 1 } { 3 ^ { 2 } } + \ldots + \frac { 1 } { n ^ { 2 } } > \int _ { 1 } ^ { n + 1 } \frac { 1 } { x ^ { 2 } } \mathrm {~d} x$$
(ii) By considering the areas of another set of rectangles, explain why
$$\frac { 1 } { 2 ^ { 2 } } + \frac { 1 } { 3 ^ { 2 } } + \frac { 1 } { 4 ^ { 2 } } + \ldots + \frac { 1 } { n ^ { 2 } } < \int _ { 1 } ^ { n } \frac { 1 } { x ^ { 2 } } \mathrm {~d} x$$
(iii) Hence show that
$$1 - \frac { 1 } { n + 1 } < \sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } } < 2 - \frac { 1 } { n }$$
(iv) Hence give bounds between which $\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } }$ lies.
\hfill \mbox{\textit{OCR FP2 2007 Q6 [11]}}