Expand and simplify \((r+1)^{4}-r^{4}\).
Use the method of differences together with the standard results for \(\sum_{r=1}^{n} r\) and \(\sum_{r=1}^{n} r^{2}\) to show that
\(\sum_{r=1}^{n} r^{3}=\frac{1}{4} n^{2}(n+1)^{2} .\)
Use the List of Formulae (MF10) to show that \(\sum_{r=1}^{13}\left(3 r^{2}-5 r+1\right)\) and \(\sum_{r=0}^{9}\left(r^{3}-1\right)\) have the same numerical value.
Use the method of differences to find \(\sum_{r=1}^{n} \frac{1}{(2 r)^{2}-1}\).
Deduce the value of \(\sum_{r=1}^{\infty} \frac{1}{(2 r)^{2}-1}\).
Verify that \(\dfrac{1}{(3r+1)(3r+4)}=\dfrac{1}{3}\left(\dfrac{1}{3r+1}-\dfrac{1}{3r+4}\right)\).
Let \(S_N\) denote \(\sum_{r=1}^{N}\dfrac{1}{(3r+1)(3r+4)}\) and let \(S\) denote \(\sum_{r=1}^{\infty}\dfrac{1}{(3r+1)(3r+4)}\). Find the least value of \(N\) such that \(S-S_N\lt\dfrac{1}{10000}\).
Express \(\frac{4}{r(r+1)(r+2)}\) in partial fractions and hence find \(\sum_{r=1}^{n} \frac{4}{r(r+1)(r+2)}\).
Deduce the value of \(\sum_{r=1}^{\infty} \frac{4}{r(r+1)(r+2)}\).
Find \(\sum_{r=1}^{n}(4 r-3)(4 r+1)\), giving your answer in its simplest form.
Given that
\(u_{k}=\frac{1}{\sqrt{2 k-1}}-\frac{1}{\sqrt{2 k+1}},\)
express \(\sum_{k=13}^{n} u_{k}\) in terms of \(n\).
Deduce the value of \(\sum_{k=13}^{\infty} u_{k}\).
Given that
\(u_{k}=\frac{1}{\sqrt{2 k-1}}-\frac{1}{\sqrt{2 k+1}},\)
express \(\sum_{k=13}^{n} u_{k}\) in terms of \(n\).
Deduce the value of \(\sum_{k=13}^{\infty} u_{k}\).
Use the method of differences to show that \(\sum_{r=1}^{N} \frac{1}{(2 r+1)(2 r+3)}=\frac{1}{6}-\frac{1}{2(2 N+3)}\).
Deduce that \(\sum_{r=N+1}^{2 N} \frac{1}{(2 r+1)(2 r+3)}\lt \frac{1}{8 N}\).
Let \(\mathrm{f}(r)=r!(r-1)\). Simplify \(\mathrm{f}(r+1)-\mathrm{f}(r)\) and hence find \(\sum_{r=n+1}^{2 n} r!\left(r^{2}+1\right)\).
It is given that
\(S_{n}=\sum_{r=1}^{n} u_{r}=2 n^{2}+n\)
Write down the values of \(S_{1}, S_{2}, S_{3}, S_{4}\). Express \(u_{r}\) in terms of \(r\), justifying your answer.
Find
\(\sum_{r=n+1}^{2 n} u_{r} .\)
It is given that
\(S_{n}=\sum_{r=1}^{n} u_{r}=2 n^{2}+n\)
Write down the values of \(S_{1}, S_{2}, S_{3}, S_{4}\). Express \(u_{r}\) in terms of \(r\), justifying your answer.
Find
\(\sum_{r=n+1}^{2 n} u_{r} .\)
Express \(\frac{1}{r(r+1)(r-1)}\) in partial fractions.
Find
\(\sum_{r=2}^{n} \frac{1}{r(r+1)(r-1)}\)
State the value of
\(\sum_{r=2}^{\infty} \frac{1}{r(r+1)(r-1)} .\)
Use induction to prove that
\(\sum_{n=1}^{N} \frac{4 n+1}{n(n+1)(2 n-1)(2 n+1)}=1-\frac{1}{(N+1)(2 N+1)}\)
Show that
\(\sum_{n=N+1}^{2 N} \frac{4 n+1}{n(n+1)(2 n-1)(2 n+1)}\lt \frac{3}{8 N^{2}} .\)
Express \(\frac{1}{(2 r+1)(2 r+3)}\) in partial fractions and hence use the method of differences to
\(\sum_{r=1}^{n} \frac{1}{(2 r+1)(2 r+3)}\)
Deduce the value of
\(\sum_{r=1}^{\infty} \frac{1}{(2 r+1)(2 r+3)}\)
Expand and simplify \((r+1)^{4}-r^{4}\).
Use the method of differences together with the standard results for \(\sum_{r=1}^{n} r\) and \(\sum_{r=1}^{n} r^{2}\) to show that
\(\sum_{r=1}^{n} r^{3}=\frac{1}{4} n^{2}(n+1)^{2} .\)
Show that \(\sum_{r=n+1}^{2 n} r^{2}=\frac{1}{6} n(2 n+1)(7 n+1)\).
The sum \(S_{N}\) is defined by \(S_{N}=\sum_{n=1}^{N} n^{5}\). Using the identity
\(\left(n+\frac{1}{2}\right)^{6}-\left(n-\frac{1}{2}\right)^{6} \equiv 6 n^{5}+5 n^{3}+\frac{3}{8} n\)
find \(S_{N}\) in terms of \(N\). [You need not simplify your result.]
Hence find \(\lim _{N \rightarrow \infty} N^{-\lambda} S_{N}\), for each of the two cases
(i) \(\lambda=6\),
(ii) \(\lambda\gt 6\).
Find \(2^{2}+4^{2}+\ldots+(2 n)^{2}\).
Hence find \(1^{2}-2^{2}+3^{2}-4^{2}+\ldots-(2 n)^{2}\), simplifying your answer.
\(\mathbf{1}\) Verify that \(\frac{1}{n^{2}}-\frac{1}{(n+1)^{2}}=\frac{2 n+1}{n^{2}(n+1)^{2}}\).
Let \(S_{N}=\sum_{r=1}^{N} \frac{2 r+1}{r^{2}(r+1)^{2}}\). Express \(S_{N}\) in terms of \(N\).
Let \(S=\lim _{N \rightarrow \infty} S_{N}\). Find the least value of \(N\) such that \(S-S_{N}\lt 10^{-16}\).