Questions

1. Question 1

   Expand the following upto 4 terms, taking the values of $x$ such that the expansion in each case is valid:

   (i)

   $(1+x)^{-1/3}$

   (ii)

   $(4-3x)^{1/2}$

   (iii)

   $\dfrac{(1-x)^{-1}}{(1+x)^2}$

   (iv)

   $\dfrac{\sqrt{1+2x}}{1-x}$

   SolutionTheory
2. Question 2

   Find the coefficient of $x^n$ in the expansion of:

   (i)

   $\dfrac{1+x^2}{(1+x)^2}$

   (ii)

   $\dfrac{(1+x)^2}{(1-x)^2}$

   SolutionTheory
3. Question 3

   If $x$ is so small that its square and higher powers can be neglected, then show that:

   (i)

   $\dfrac{1-x}{\sqrt{1+x}} \approx 1 - \dfrac{3}{2}x$

   (ii)

   $\dfrac{\sqrt{1+2x}}{\sqrt{1-x}} \approx 1 + \dfrac{3}{2}x$

   (iii)

   $\dfrac{(9+7x)^{1/2} - (16+3x)^{1/4}}{4+5x} \approx \dfrac{1}{4} - \dfrac{17}{384}x$

   (iv)

   $\dfrac{\sqrt{4+x}}{(1-x)^3} \approx 2 + \dfrac{25}{4}x$

   SolutionTheory
4. Question 4

   If $x$ is so small that its cube and higher power can be neglected, show that:

   (i)

   $\sqrt{1-x-2x^2} \approx 1 - \dfrac{1}{2}x - \dfrac{9}{8}x^2$

   (ii)

   $\sqrt{\dfrac{1+x}{1-x}} \approx 1 + x + \dfrac{1}{2}x^2$

   SolutionTheory
5. Question 5

   If $x$ is very nearly equal to 1, then prove that $px^p - qx^q \approx (p-q)x^{p+q}$.

   SolutionTheory
6. Question 6

   Identify the following series as binomial expansion and find the sum. $1 - \dfrac{1}{2}\left(\dfrac{1}{4}\right) + \dfrac{1\cdot 3}{2!}\left(\dfrac{1}{4}\right)^2 - \dfrac{1\cdot 3\cdot 5}{3!}\left(\dfrac{1}{4}\right)^3 + \cdots$

   SolutionTheory
7. Question 7

   Use binomial theorem to show that $1 + \dfrac{1}{4} + \dfrac{1\cdot 3}{4\cdot 8} + \dfrac{1\cdot 3\cdot 5}{4\cdot 8\cdot 12} + \cdots = \sqrt{2}$.

   SolutionTheory
8. Question 8

   If $y = \dfrac{1}{3} + \dfrac{1\cdot 3}{2!}\left(\dfrac{1}{3}\right)^2 + \dfrac{1\cdot 3\cdot 5}{3!}\left(\dfrac{1}{3}\right)^3 + \cdots$ prove that $y^2 + 2y - 2 = 0$.

   SolutionTheory
9. Question 9

   If $2y = \dfrac{1}{2^2} + \dfrac{1\cdot 3}{2!}\cdot \dfrac{1}{2^4} + \dfrac{1\cdot 3\cdot 5}{3!}\cdot \dfrac{1}{2^6} + \cdots$, prove that $4y^2 + 4y - 1 = 0$.

   SolutionTheory
10. Question 10

    Show that the coefficient of $x^r$ in $\dfrac{x}{(1-px)(1-qx)}$ is $\dfrac{p^r - q^r}{p-q}$.

    SolutionTheory