June 2023 Paper 3 Q11

OCR MEICurrent spec3 marksIntegration Sequences & Series

The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.

The relevant parts of the article “Approximating series” are reproduced below; the line numbers are those printed on the Insert.

Line 10
Euler’s approximate summation formula
Lines 11–13
In 1741, the mathematician Leonhard Euler published an approximate formula for summing a series. In modern notation, this can be expressed as follows.
\(\displaystyle\sum_{r=1}^{n}\mathrm{f}(r) \approx \int_1^n \mathrm{f}(x)\,\mathrm{d}x + \frac{\mathrm{f}(n) + \mathrm{f}(1)}{2} + \frac{\mathrm{f}(1) - \mathrm{f}(2)}{12} - \frac{\mathrm{f}(n) - \mathrm{f}(n+1)}{12}\)

(a) Evaluate \(\displaystyle\sum_{r=1}^{5} r^2\). [1]

(b) Show that Euler’s approximate formula, as given in line 13, gives the exact value of \(\displaystyle\sum_{r=1}^{5} r^2\). [2]

Scheme	Marks	AO
55	B1	1.1
	[1]

Mark scheme (b)

Scheme	Marks	AO
\(\displaystyle\int_1^5 x^2\,\mathrm{d}x + \frac{5^2 + 1^2}{2} + \frac{1^2 - 2^2}{12} - \frac{5^2 - 6^2}{12}\)	M1	1.1
\(\left[\dfrac{x^3}{3}\right]_1^5 + \dfrac{26}{2} - \dfrac{3}{12} + \dfrac{11}{12} = \dfrac{124}{3} + \dfrac{41}{3} = 55\)	A1	2.2a
	[2]

Notes

M1: Correct substitution into formula
Condone \(r\) but not \(n\) instead of \(x\)
Square numbers may be evaluated

A1: Correct completion.
Integration may be done BC
At least one step to be shown