| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Integral bounds for series |
| Difficulty | Challenging +1.8 This is a sophisticated Further Maths question requiring understanding of integral bounds for series, manipulation of the harmonic series, and Maclaurin series approximation. Part (i) requires geometric insight about rectangles and integrals; part (ii) involves algebraic manipulation to find bounds; part (iii) requires Taylor series expansion and asymptotic analysis. While the individual techniques are standard for FP2, the combination and the conceptual understanding of using integrals to bound series makes this significantly harder than average A-level questions, though not exceptionally difficult for Further Maths students who have practiced these topics. |
| Spec | 1.08d Evaluate definite integrals: between limits4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(a=2,\ b=n\) | B1 | For any 2 correct |
| \(c=1,\ d=n-1\) | B1 | For the third correct |
| B1 | For all four correct. Allow values inserted in series. SC treat \(a=\frac{1}{2}\) etc as MR−1 once |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int_1^n \frac{1}{x} \, dx = \ln n\) | B1 | For integral evaluated soi (Definite integral between 1 and \(n\)) |
| \(1 + \frac{1}{2} + \ldots + \frac{1}{n} < 1 + \ln n\) | M1 | For adding \(1\) OR \(\frac{1}{n}\) to series |
| \(\Rightarrow f(n) < 1\) (upper bound) | A1 | For correct upper bound |
| \(\Rightarrow f(n) > \frac{1}{n}\) (lower bound) | A1 | For correct lower bound |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(f(n+1) - f(n) = \frac{1}{n+1} - \ln(n+1) + \ln n\) | B1 | For correct expression |
| \(= \frac{1}{n+1} - \ln\!\left(\frac{n+1}{n}\right) \approx \frac{1}{n+1} - \left(\frac{1}{n} - \frac{1}{2n^2}\right)\) | M1 | For combining ln terms |
| M1 | For attempt to expand \(\ln\!\left(1+\frac{1}{n}\right)\) | |
| A1 | Correct expansion of \(\ln\!\left(1+\frac{1}{n}\right)\) | |
| \(\approx \frac{1}{n+1} - \frac{2n-1}{2n^2}\) | ||
| \(\approx -\dfrac{n-1}{2n^2(n+1)}\) | A1 | For correct expression AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(f(n+1)-f(n) = \frac{1}{n+1} - \ln(n+1) + \ln n\) | B1 | For correct expression |
| \(= \frac{1}{n+1} + \ln\!\left(\frac{n}{n+1}\right) = \frac{1}{n+1} + \ln\!\left(1 - \frac{1}{n+1}\right)\) | M1 | For combining ln terms and attempt to expand |
| \(= \frac{1}{n+1} + \left(-\frac{1}{(n+1)} - \frac{1}{2(n+1)^2}\right)\) | M1 | For attempt to expand \(\ln\!\left(1-\frac{1}{(n+1)}\right)\) |
| A1 | Correct expansion of \(\ln\!\left(1-\frac{1}{(n+1)}\right)\) | |
| \(= -\dfrac{1}{2(n+1)^2}\) |
## Question 7:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a=2,\ b=n$ | B1 | For any 2 correct |
| $c=1,\ d=n-1$ | B1 | For the third correct |
| | B1 | For all four correct. Allow values inserted in series. SC treat $a=\frac{1}{2}$ etc as MR−1 once |
**[3]**
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int_1^n \frac{1}{x} \, dx = \ln n$ | B1 | For integral evaluated **soi** (Definite integral between 1 and $n$) |
| $1 + \frac{1}{2} + \ldots + \frac{1}{n} < 1 + \ln n$ | M1 | For adding $1$ OR $\frac{1}{n}$ to series |
| $\Rightarrow f(n) < 1$ (upper bound) | A1 | For correct upper bound |
| $\Rightarrow f(n) > \frac{1}{n}$ (lower bound) | A1 | For correct lower bound |
**[4]**
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $f(n+1) - f(n) = \frac{1}{n+1} - \ln(n+1) + \ln n$ | B1 | For correct expression |
| $= \frac{1}{n+1} - \ln\!\left(\frac{n+1}{n}\right) \approx \frac{1}{n+1} - \left(\frac{1}{n} - \frac{1}{2n^2}\right)$ | M1 | For combining ln terms |
| | M1 | For attempt to expand $\ln\!\left(1+\frac{1}{n}\right)$ |
| | A1 | Correct expansion of $\ln\!\left(1+\frac{1}{n}\right)$ |
| $\approx \frac{1}{n+1} - \frac{2n-1}{2n^2}$ | | |
| $\approx -\dfrac{n-1}{2n^2(n+1)}$ | A1 | For correct expression **AG** |
**[5]**
**Alternative for 7(iii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $f(n+1)-f(n) = \frac{1}{n+1} - \ln(n+1) + \ln n$ | B1 | For correct expression |
| $= \frac{1}{n+1} + \ln\!\left(\frac{n}{n+1}\right) = \frac{1}{n+1} + \ln\!\left(1 - \frac{1}{n+1}\right)$ | M1 | For combining ln terms and attempt to expand |
| $= \frac{1}{n+1} + \left(-\frac{1}{(n+1)} - \frac{1}{2(n+1)^2}\right)$ | M1 | For attempt to expand $\ln\!\left(1-\frac{1}{(n+1)}\right)$ |
| | A1 | Correct expansion of $\ln\!\left(1-\frac{1}{(n+1)}\right)$ |
| $= -\dfrac{1}{2(n+1)^2}$ | | |
**Max 4**
---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
$$\int _ { 1 } ^ { n } \frac { 1 } { x } \mathrm {~d} x < \frac { 1 } { c } + \frac { 1 } { c + 1 } + \frac { 1 } { c + 2 } + \ldots + \frac { 1 } { d }$$
(i) 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 $n$.\\
(ii) Use your answers to part (i) to obtain upper and lower bounds for $\mathrm { f } ( n )$.\\
(iii) By using the first 2 terms of the Maclaurin series for $\ln ( 1 + x )$ show that, for large $n$,
$$f ( n + 1 ) - f ( n ) \approx - \frac { n - 1 } { 2 n ^ { 2 } ( n + 1 ) } .$$
\hfill \mbox{\textit{OCR FP2 2012 Q7 [12]}}