Pure Mathematics Practice: Series and Sequences — Answers

Questions: Series and Sequences Drills.

Question 1

(a)

$$u_{20}=7+19(3)=64.$$

(b)

$$S_{20}=\frac{20}{2}(7+64)=710.$$

(c) Since $|0.8|<1$, the geometric series converges and

$$S_{\infty }=\frac{5}{1-0.8}=25.$$

Question 2

Using the standard sums,

$$\begin{array}{rl}\sum _{r=1}^n(3r^2-2r)& =3\frac{n(n+1)(2n+1)}{6}-2\frac{n(n+1)}{2}\\ & =\frac{n(n+1)(2n-1)}{2}.\end{array}$$

Question 3

The common ratio is $2k-1$. Convergence requires

$$|2k-1|<1,$$

so

$$0<k<1.$$

For these values,

$$S_{\infty }=\frac{1}{1-(2k-1)}=\frac{1}{2(1-k)}.$$

Question 4

(a)

$$e^{2x}=1+2x+2x^2+\frac{4}{3}{x}^3+O(x^4)$$

and

$$\cos x=1-\frac{1}{2}{x}^2+O(x^4).$$

Multiplying and retaining terms through $x^3$ gives

$$e^{2x}\cos x=1+2x+\frac{3}{2}{x}^2+\frac{1}{3}{x}^3+O(x^4).$$

(b) Put $x=0.1$:

$$e^{0.2}\cos (0.1)\approx 1+0.2+\frac{3}{2}(0.1{)}^2+\frac{1}{3}(0.1{)}^3=1.215333\dots .$$

Hence the estimate is

$$1.21533$$

to 5 decimal places.

The omitted error is of order $(0.1{)}^4$, so the expansion supports a close approximation, although the order notation alone does not guarantee every displayed decimal without an error bound.