3 (a) Use the method of differences to find \(\sum_{r=1}^{n} \frac{1}{(kr+1)(kr-k+1)}\) in terms of \(n\) and \(k\), where \(k\) is a positive constant.
(b) Deduce the value of \(\sum_{r=1}^{\infty} \frac{1}{(kr+1)(kr-k+1)}\).
(c) Find also \(\sum_{r=n}^{n^2} \frac{1}{(kr+1)(kr-k+1)}\) in terms of \(n\) and \(k\).
The cubic equation \(x^3 + 4x^2 + 6x + 1 = 0\) has roots \(\alpha, \beta, \gamma\).
(a) Find the value of \(\alpha^2 + \beta^2 + \gamma^2\).
(b) Use standard results from the list of formulae (MF19) to show that
\(\sum_{r=1}^{n} ((\alpha + r)^2 + (\beta + r)^2 + (\gamma + r)^2) = n(n^2 + an + b),\)
where \(a\) and \(b\) are constants to be determined.
3 (a) Use the method of differences to find \(\sum_{r=1}^{n} \frac{1}{(kr+1)(kr-k+1)}\) in terms of \(n\) and \(k\), where \(k\) is a positive constant.
(b) Deduce the value of \(\sum_{r=1}^{\infty} \frac{1}{(kr+1)(kr-k+1)}\).
(c) Find also \(\sum_{r=n}^{n^2} \frac{1}{(kr+1)(kr-k+1)}\) in terms of \(n\) and \(k\).
2 (a) Use standard results from the list of formulae (MF19) to show that
\(\sum_{r=1}^{n} (6r^2 + 6r - 5) = an^3 + bn^2 + cn\),
where \(a, b\) and \(c\) are integers to be determined.
(b) Use the method of differences to find \(\sum_{r=1}^{n} \frac{6r^2 + 6r - 5}{r^2 + r}\) in terms of \(n\).
(c) Find also \(\sum_{r=n+1}^{2n} \frac{6r^2 + 6r - 5}{r^2 + r}\) in terms of \(n\).
Let \(a\) be a positive constant.
(a) Use the method of differences to find \(\sum_{r=1}^{n} \frac{1}{(ar+1)(ar+a+1)}\) in terms of \(n\) and \(a\).
(b) Find the value of \(a\) for which \(\sum_{r=1}^{\infty} \frac{1}{(ar+1)(ar+a+1)} = \frac{1}{6}\).
(a) By considering \((2r+1)^3 - (2r-1)^3\), use the method of differences to prove that \(\sum_{r=1}^{n} r^2 = \frac{1}{6}n(n+1)(2n+1)\).
Let \(S_n = 1^2 + 3 \times 2^2 + 3^2 + 3 \times 4^2 + 5^2 + 3 \times 6^2 + \ldots + \left(2 + (-1)^n\right)n^2\).
(b) Show that \(S_{2n} = \frac{1}{3}n(2n+1)(an+b)\), where \(a\) and \(b\) are integers to be determined.
(c) State the value of \(\lim_{n \to \infty} \frac{S_{2n}}{n^3}\).
(a) Use the list of formulae (MF19) to find \(\sum_{r=1}^{n} r(r+2)\) in terms of \(n\), simplifying your answer.
(b) Express \(\frac{1}{r(r+2)}\) in partial fractions and hence find \(\sum_{r=1}^{n} \frac{1}{r(r+2)}\) in terms of \(n\).
(c) Deduce the value of \(\sum_{r=1}^{\infty} \frac{1}{r(r+2)}\).
(a) Use standard results from the List of formulae (MF19) to find \(\sum_{r=1}^{n} (1 - r - r^2)\) in terms of \(n\), simplifying your answer.
(b) Show that \(\frac{1 - r - r^2}{(r^2 + 2r + 2)(r^2 + 1)} = \frac{r + 1}{(r+1)^2 + 1} - \frac{r}{r^2 + 1}\) and hence use the method of differences to find \(\sum_{r=1}^{n} \frac{1 - r - r^2}{(r^2 + 2r + 2)(r^2 + 1)}\).
(a) Use standard results from the list of formulae (MF19) to find \(\sum_{r=1}^{n} r(r+1)(r+2)\) in terms of \(n\), fully factorising your answer.
(b) Express \(\frac{1}{r(r+1)(r+2)}\) in partial fractions and hence use the method of differences to find \(\sum_{r=1}^{n} \frac{1}{r(r+1)(r+2)}\).
(c) Deduce the value of \(\sum_{r=1}^{\infty} \frac{1}{r(r+1)(r+2)}\).
The cubic equation \(x^3 + 2x^2 + 3x + 3 = 0\) has roots \(\alpha, \beta, \gamma\).
(a) Find the value of \(\alpha^2 + \beta^2 + \gamma^2\).
(b) Show that \(\alpha^3 + \beta^3 + \gamma^3 = 1\).
(c) Use standard results from the list of formulae (MF19) to show that
\(\sum_{r=1}^{n} ((\alpha + r)^3 + (\beta + r)^3 + (\gamma + r)^3) = n + \frac{1}{4}n(n+1)(an^2 + bn + c),\)
where \(a, b, c\) are constants to be determined.
2 (a) Use standard results from the List of Formulae (MF19) to show that
\(\sum_{r=1}^{n}(7 r+1)(7 r+8)=a n^{3}+b n^{2}+c n\)
where \(a, b\) and \(c\) are constants to be determined.
(b) Use the method of differences to find \(\sum_{r=1}^{n} \frac{1}{(7 r+1)(7 r+8)}\) in terms of \(n\).
(c) Deduce the value of \(\sum_{r=1}^{\infty} \frac{1}{(7 r+1)(7 r+8)}\).
4 (a) By first expressing \(\frac{1}{r^{2}-1}\) in partial fractions, show that
\(\sum_{r=2}^{n} \frac{1}{r^{2}-1}=\frac{3}{4}-\frac{a n+b}{2 n(n+1)},\)
where \(a\) and \(b\) are integers to be found.
(b) Deduce the value of \(\sum_{r=2}^{\infty} \frac{1}{r^{2}-1}\).
(c) Find the limit, as \(n \rightarrow \infty\), of \(\sum_{r=n+1}^{2 n} \frac{n}{r^{2}-1}\).
4 (i) Use the method of differences to show that \(\sum_{r=1}^{N} \frac{1}{(3 r+1)(3 r-2)}=\frac{1}{3}-\frac{1}{3(3 N+1)}\).
(ii) Find the limit, as \(N \rightarrow \infty\), of \(\sum_{r=N+1}^{N^{2}} \frac{N}{(3 r+1)(3 r-2)}\).
\(5 \quad\) Let \(S_{N}=\sum_{r=1}^{N}(5 r+1)(5 r+6)\) and \(T_{N}=\sum_{r=1}^{N} \frac{1}{(5 r+1)(5 r+6)}\).
(i) Use standard results from the List of Formulae (MF10) to show that
\(S_{N}=\frac{1}{3} N\left(25 N^{2}+90 N+83\right)\)
(ii) Use the method of differences to express \(T_{N}\) in terms of \(N\).
monisinatin
(iii) Find \(\lim _{N \rightarrow \infty}\left(N^{-3} S_{N} T_{N}\right)\).
Let \(S_{n}=\sum_{r=1}^{n}(-1)^{r-1} r^{2}\).
(i) Use the standard result for \(\sum_{r=1}^{n} r^{2}\) given in the List of Formulae (MF10) to show that
\(S_{2 n}=-n(2 n+1) .\)
(ii) State the value of \(\lim _{n \rightarrow \infty} \frac{S_{2 n}}{n^{2}}\) and find \(\lim _{n \rightarrow \infty} \frac{S_{2 n+1}}{n^{2}}\).
Answer only one of the following two alternatives.
EITHER
(i) Show that if \(z=e^{i\theta}\) and \(z\neq -1\), then \(\frac{z-1}{z+1}=i\tan\frac{\theta}{2}\).
(ii) Hence, or otherwise, show that if \(z\) is a cube root of unity, then
\(\frac{z^3-1}{z^3+1}+\frac{z^2-1}{z^2+1}+\frac{z-1}{z+1}=0.\)
(iii) Hence write down three roots of
\((z^3-1)(z^2+1)(z+1)+(z^2-1)(z^3+1)(z+1)+(z-1)(z^3+1)(z^2+1)=0\)
and find the other three roots. Give your answers in an exact form.
OR
It is given that \(\mathbf e\) is an eigenvector of \(\mathbf A\), with corresponding eigenvalue \(\lambda\).
(i) Write down another eigenvector of \(\mathbf A\) corresponding to \(\lambda\).
(ii) Write down an eigenvector and corresponding eigenvalue of \(\mathbf A^n\), where \(n\) is a positive integer.
Let \(\mathbf A=\begin{pmatrix}3&0&0\\2&7&0\\4&8&1\end{pmatrix}\).
(iii) Find a matrix \(\mathbf P\) and a diagonal matrix \(\mathbf D\) such that \(\mathbf A^n=\mathbf P\mathbf D\mathbf P^{-1}\).
(iv) Determine the set of values of the real constant \(k\) such that \(\sum_{n=1}^{\infty} k^n(\mathbf A^n-k\mathbf A^{n+1})=k\mathbf A\).
(i) Verify that
\(\frac{n(\mathrm{e}-1)+\mathrm{e}}{n(n+1) \mathrm{e}^{n+1}}=\frac{1}{n \mathrm{e}^{n}}-\frac{1}{(n+1) \mathrm{e}^{n+1}} .\)
Let \(S_{N}=\sum_{n=1}^{N} \frac{n(\mathrm{e}-1)+\mathrm{e}}{n(n+1) \mathrm{e}^{n+1}}\).
(ii) Express \(S_{N}\) in terms of \(N\) and e.
Let \(S=\lim _{N \rightarrow \infty} S_{N}\).
(iii) Find the least value of \(N\) such that \((N+1)\left(S-S_{N}\right)\lt10^{-3}\).
Let
\(S_{N}=\sum_{r=1}^{N}(3 r+1)(3 r+4) \quad \text { and } \quad T_{N}=\sum_{r=1}^{N} \frac{1}{(3 r+1)(3 r+4)} .\)
(i) Use standard results from the List of Formulae (MF10) to show that
\(S_{N}=N\left(3 N^{2}+12 N+13\right)\)
(ii) Use the method of differences to show that
\(T_{N}=\frac{1}{12}-\frac{1}{3(3 N+4)}\)
(iii) Deduce that \(\frac{S_{N}}{T_{N}}\) is an integer.
(iv) Find \(\lim _{N \rightarrow \infty} \frac{S_{N}}{N^{3} T_{N}}\).
It is given that \(\sum_{r=1}^{n} u_{r}=n^{2}(2 n+3)\), where \(n\) is a positive integer.
(i) Find \(\sum_{r=n+1}^{2 n} u_{r}\).
(ii) Find \(u_{r}\).
(i) Verify that \(\frac{2 r+1}{r(r+1)(r+2)}=\frac{1}{2}\left\{\frac{(2 r+1)(2 r+3)}{(r+1)(r+2)}-\frac{(2 r-1)(2 r+1)}{r(r+1)}\right\}\).
(ii) Hence show that \(\sum_{r=1}^{n} \frac{2 r+1}{r(r+1)(r+2)}=\frac{1}{2}\left\{\frac{(2 n+1)(2 n+3)}{(n+1)(n+2)}-\frac{3}{2}\right\}\).
(iii) Deduce the value of \(\sum_{r=1}^{\infty} \frac{2 r+1}{r(r+1)(r+2)}\).