FP2: SeriesEdexcel International A Level Further Maths: Topic test

1. 1
   A series has general term $u_r=\frac{1}{r(r+1)}$ for $r\ge1$.

   (a)

   Which of the following is $u_r$ written in partial fractions?

   [1 mark]

   • A$\frac1r-\frac1{r+1}$
   • B$\frac1r+\frac1{r+1}$
   • C$\frac1{r+1}-\frac1r$
   • D$\frac12\left(\frac1r-\frac1{r+1}\right)$

   (b)

   Find $\sum_{r=1}^{n}u_r$.

   [1 mark]

   • A$\frac{1}{n+1}$
   • B$1+\frac{1}{n+1}$
   • C$\frac{n}{n+1}$
   • D$1-\frac1n$

   (c)

   Find the sum to infinity of the series.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $f(r)=2^{r}$ for positive integers $r$.

   (a)

   Find $f(r+1)-f(r)$.

   [1 mark]

   • A$1$
   • B$2^{r}$
   • C$2^{r+1}$
   • D$2$

   (b)

   Find $\sum_{r=1}^{n}2^{r}$.

   [1 mark]

   • A$2^{n+1}-1$
   • B$2^{n}-2$
   • C$2^{n+1}$
   • D$2^{n+1}-2$

   (c)

   Hence find $\sum_{r=4}^{10}2^{r}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A series has general term $u_r=\frac{1}{(3r-1)(3r+2)}$ for $r\ge1$.

   (a)

   Express $u_r$ in partial fractions.

   [3 marks]

   (b)

   Hence find $\sum_{r=1}^{n}u_r$, giving your answer as a single fraction in terms of $n$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The general term of a series is $u_r=\frac{r}{(r+1)!}$ for $r\ge1$.

   (a)

   Show that $u_r=\frac{1}{r!}-\frac{1}{(r+1)!}$ and hence find $\sum_{r=1}^{n}u_r$ in terms of $n$.

   [6 marks]

   (b)

   Find the least value of $n$ for which $\sum_{r=3}^{n}u_r$ is within $0.001$ of $\sum_{r=3}^{\infty}u_r$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Let $f(r)=r^3$ for positive integers $r$.

   (a)

   Find $f(r)-f(r-1)$.

   [1 mark]

   • A$1$
   • B$3r^2+3r+1$
   • C$3r^2-1$
   • D$3r^2-3r+1$

   (b)

   Find $\sum_{r=1}^{n}\left(3r^2-3r+1\right)$.

   [1 mark]

   • A$n^3$
   • B$n^3-1$
   • C$n^3+1$
   • D$(n+1)^3$

   (c)

   Hence find $\sum_{r=6}^{10}\left(3r^2-3r+1\right)$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Let $f(r)=\frac{1}{r^2}$ for positive integers $r$.

   (a)

   Find $f(r)-f(r+1)$ as a single fraction.

   [1 mark]

   • A$\frac{1}{r^2(r+1)^2}$
   • B$\frac{2r-1}{r^2(r+1)^2}$
   • C$\frac{2r+1}{r^2(r+1)^2}$
   • D$-\frac{2r+1}{r^2(r+1)^2}$

   (b)

   Find $\sum_{r=1}^{n}\frac{2r+1}{r^2(r+1)^2}$.

   [1 mark]

   • A$1-\frac{1}{n^2}$
   • B$1-\frac{1}{(n+1)^2}$
   • C$\frac{1}{(n+1)^2}-1$
   • D$1+\frac{1}{(n+1)^2}$

   (c)

   Hence evaluate $\sum_{r=1}^{9}\frac{2r+1}{r^2(r+1)^2}$ exactly.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A series has general term $u_r=\ln\left(\frac{r(r+2)}{(r+1)^2}\right)$ for $r\ge1$.

   (a)

   Show that $u_r=f(r)-f(r+1)$, where $f(r)=\ln\left(\frac{r}{r+1}\right)$.

   [3 marks]

   (b)

   Hence find $\sum_{r=1}^{n}u_r$, and state the sum to infinity.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A series has general term $u_r=\frac{2r+3}{r(r+1)(r+2)}$ for $r\ge1$.

   (a)

   Express $u_r$ in partial fractions, and hence show that $\sum_{r=1}^{n}u_r=\frac74-\frac{3}{2(n+1)}-\frac{1}{2(n+2)}$.

   [6 marks]

   (b)

   Find the sum to infinity of the series, and the least value of $n$ for which $\sum_{r=1}^{n}u_r$ differs from the sum to infinity by less than $0.01$.

   [6 marks]

   Total for question 8: 12 marks

End of questions