4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n

Let \(C = \sum_{r=0}^{20} \binom{20}{r} \cos r\theta\). Show that \(C = 2^{20} \cos^{20}\left(\frac{1}{2}\theta\right) \cos 10\theta\). [8]

The function \(f\) is defined by \(f(x) = \cosh\left(\frac{x}{2}\right)\).

1. State the Maclaurin series expansion for \(\cosh\left(\frac{x}{2}\right)\) up to and including the term in \(x^4\). [2]

Another function \(g\) is defined by \(g(x) = x^2 - 2\). The diagram below shows parts of the graphs of \(y = f(x)\) and \(y = g(x)\). \includegraphics{figure_2}

1. The two graphs intersect at the point A, as shown in the diagram. Use your answer from part (a) to find an approximation for the \(x\)-coordinate of A, giving your answer correct to two decimal places. [5]
2. Using your answer to part (b), find an approximation for the area of the shaded region enclosed by the two graphs, the \(x\)-axis and the \(y\)-axis. [6]

The function \(f\) is given by $$f(x) = e^x \cos x.$$

1. Show that \(f''(x) = -2e^x \sin x\). [2]
2. Determine the Maclaurin series for \(f(x)\) as far as the \(x^4\) term. [6]
3. Hence, by differentiating your series, determine the Maclaurin series for \(e^x \sin x\) as far as the \(x^3\) term. [4]
4. The equation $$10e^x \sin x - 11x = 0$$ has a small positive root. Determine its approximate value, giving your answer correct to three decimal places. [4]

In this question you must show detailed reasoning.

1. By using an appropriate Maclaurin series prove that if \(x > 0\) then \(e^x > 1 + x\). [2]
2. Hence, by using a suitable substitution, deduce that \(e^t > et\) for \(t > 1\). [1]
3. Using the inequality in part (b), and by making a suitable choice for \(t\), determine which is greater, \(e^{\pi}\) or \(\pi^e\). [3]

In this question you must show all stages of your working. The function \(f\) is defined by \(f(x) = (1 + 2x)^{\frac{1}{2}}\).

1. Find \(f'''(x)\) (i.e. the third derivative of \(f\)) showing all your intermediate steps. Hence, find the Maclaurin series for \(f(x)\) up to and including the \(x^3\) term. [8 marks]
2. Use the expansion of \(e^x\) together with the result in part (a) to show that, up to and including the \(x^3\) term, $$e^x(1 + 2x)^{\frac{1}{2}} = 1 + 2x + x^2 + kx^3,$$ where \(k\) is a rational number to be found. [3 marks]

Let $$f(x) = \frac{27x^2 + 32x + 16}{(3x + 2)^2(1 - x)}$$

1. Express \(f(x)\) in terms of partial fractions [5]
2. Hence, or otherwise, find the series expansion of \(f(x)\), in ascending powers of \(x\), up to and including the term in \(x^2\). Simplify each term. [6]
3. State, with a reason, whether your series expansion in part (b) is valid for \(x = \frac{1}{2}\). [2]

You are given that \(f(x) = \ln(2 + x)\).

1. Determine the exact value of \(f'(0)\). [2]
2. Show that \(f''(0) = -\frac{1}{4}\). [2]
3. Hence write down the first three terms of the Maclaurin series for \(f(x)\). [3]

The Taylor series expansion, about \(x = 1\), of the function \(y\) is $$y = 1 + \sum_{n=1}^{\infty} \frac{(-2)^{n-1}(x-1)^n}{1 \times 3 \times 5 \times \ldots \times (2n-1)}.$$ Find the value of \(\frac{\text{d}^4 y}{\text{d}x^4}\) when \(x = 1\). [3]

Given that \(y = \cos\{\ln(1 + x)\}\), prove that

1. \((1 + x)\frac{\mathrm{d}y}{\mathrm{d}x} = -\sin\{\ln(1 + x)\}\), [1]
2. \((1 + x)^2 \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (1 + x)\frac{\mathrm{d}y}{\mathrm{d}x} + y = 0\). [2]

Obtain an equation relating \(\frac{\mathrm{d}^3 y}{\mathrm{d}x^3}\), \(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}\) and \(\frac{\mathrm{d}y}{\mathrm{d}x}\). [2] Hence find Maclaurin's series for \(y\), up to and including the term in \(x^3\). [4] Verify that the same result is obtained if the standard series expansions for \(\ln(1 + x)\) and \(\cos x\) are used. [3]

Given that $$(1 + x)^n = 1 + \sum_{r=1}^{\infty} \frac{n(n-1)...(n-r+1)}{1 \times 2 \times ... \times r} x^r \quad (|x| < 1, x \in \mathbb{R}, n \in \mathbb{R})$$

1. show that $$(1 - x)^{-\frac{1}{2}} = \sum_{r=0}^{\infty} \binom{2r}{r}\left(\frac{x}{4}\right)^r$$ [5]
2. show that \((9 - 4x^2)^{-\frac{1}{2}}\) can be written in the form \(\sum_{r=0}^{\infty} \binom{2r}{r} \frac{x^{2r}}{3^{r}}\) and give \(q\) in terms of \(r\). [3]
3. Find \(\sum_{r=1}^{\infty} \binom{2r}{r} \times \frac{2r}{9} \times \left(\frac{x}{3}\right)^{2r-1}\) [3]
4. Hence find the exact value of $$\sum_{r=1}^{\infty} \binom{2r}{r} \times \frac{2r\sqrt{5}}{9} \times \frac{1}{5^r}$$ giving your answer as a rational number. [2]