| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 9 |
| 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 Further Maths question requiring students to use integral comparison tests for series convergence. While the technique is non-standard for A-level and requires geometric insight about rectangles approximating areas under curves, the actual execution is straightforward: comparing ∑1/x to ∫1/x dx (which diverges) and ∑1/x² to ∫1/x² dx (which converges to 1). The calculations themselves are routine integration, making this moderately above average difficulty. |
| Spec | 1.04g Sigma notation: for sums of series1.08g Integration as limit of sum: Riemann sums4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Construct rectangles from \(x=1\) of height \(y\) to \(n\) or \(\infty\); Sum of areas \(= \frac{1}{1}+\frac{1}{2}+\ldots = \sum_1^{\infty}\frac{1}{x}\); This area is bigger than area under curve | M1 | Or sum from 1 to \(n\); condone comparison of areas for 1 to \(n\) and 1 to \(n-1\) |
| Sum and inequality stated or implied by diagram | A1 | |
| \(A = \int_1^{\infty}\frac{1}{x}\,dx = [\ln x]_1^{\infty} = \infty\) | B1 | Integral from 1 to \(n+1\) or \(\infty\); if \(n-1\) above then integral to \(n\) |
| Since Sum \(> A\), the sum is infinite | A1 | Conclusion. [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Construct rectangles from \(x=2\) to the left of height \(y\) to \(n\) or \(\infty\); Sum \(= \frac{1}{2^2}+\frac{1}{3^2}+\ldots = \sum_2^{\infty}\frac{1}{x^2}\); This area is less than area under curve | M1 | Rectangles must be under curve; may include rectangle from \(x=1\) to the left |
| Sum and inequality stated or implied by diagram | A1 | |
| \(A = \int_1^{\infty}\frac{1}{x^2}\,dx = \left[-\frac{1}{x}\right]_1^{\infty} = 0--1 = 1\) | B1 | Area under curve |
| Since Sum \(< A\): \(\frac{1}{2^2}+\frac{1}{3^2}+\ldots = \sum_2^{\infty}\frac{1}{x^2} < 1\) | ||
| \(\Rightarrow \sum_1^{\infty}\frac{1}{x^2} < 1+1\) | M1 | For adding 1 to both sides, may appear earlier |
| Upper limit \(= 2\) | A1 | [5] |
## Question 7:
### Part (i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Construct rectangles from $x=1$ of height $y$ to $n$ or $\infty$; Sum of areas $= \frac{1}{1}+\frac{1}{2}+\ldots = \sum_1^{\infty}\frac{1}{x}$; This area is bigger than area under curve | M1 | Or sum from 1 to $n$; condone comparison of areas for 1 to $n$ and 1 to $n-1$ |
| Sum and inequality stated or implied by diagram | A1 | |
| $A = \int_1^{\infty}\frac{1}{x}\,dx = [\ln x]_1^{\infty} = \infty$ | B1 | Integral from 1 to $n+1$ or $\infty$; if $n-1$ above then integral to $n$ |
| Since Sum $> A$, the sum is infinite | A1 | Conclusion. [4] |
### Part (ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Construct rectangles from $x=2$ to the left of height $y$ to $n$ or $\infty$; Sum $= \frac{1}{2^2}+\frac{1}{3^2}+\ldots = \sum_2^{\infty}\frac{1}{x^2}$; This area is less than area under curve | M1 | Rectangles must be under curve; may include rectangle from $x=1$ to the left |
| Sum and inequality stated or implied by diagram | A1 | |
| $A = \int_1^{\infty}\frac{1}{x^2}\,dx = \left[-\frac{1}{x}\right]_1^{\infty} = 0--1 = 1$ | B1 | Area under curve |
| Since Sum $< A$: $\frac{1}{2^2}+\frac{1}{3^2}+\ldots = \sum_2^{\infty}\frac{1}{x^2} < 1$ | | |
| $\Rightarrow \sum_1^{\infty}\frac{1}{x^2} < 1+1$ | M1 | For adding 1 to both sides, may appear earlier |
| Upper limit $= 2$ | A1 | [5] |
---7 (i) 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.\\
(ii) 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 $\sum _ { x = 1 } ^ { \infty } \frac { 1 } { x ^ { 2 } }$.
\hfill \mbox{\textit{OCR FP2 2016 Q7 [9]}}