Exercise Set 11

  1. Write the series \[\begin{equation*} \frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\frac{1}{4\cdot 5}+\frac{1}{5\cdot 6} \end{equation*}\] using Sigma notation in the forms \(\displaystyle\sum_{k=1}^{n}(\dots)\) and \(\displaystyle\sum_{l=0}^{m}(\dots)\).

  2. Evaluate \(\displaystyle\sum_{n=1}^{10}\frac{2}{3^n}\) and \(\displaystyle\sum_{n=0}^{9}\frac{2}{3^n}\) using the formula for the \(n^{\text{th}}\) partial sum of a geometric series.

  3. Do the following geometric series converge or diverge? If they converge, find their sum.

    1. \(\displaystyle\sum_{n=1}^{\infty}\frac{2}{3^n}\)
    2. \(\displaystyle\sum_{n=1}^{\infty}(-1)^{n-1}\frac{1}{2^{n-1}}\)
    3. \(\displaystyle\sum_{n=1}^{\infty}\left( \dfrac{\pi}{2}\right)^n\)

  4. Find the \(3^{\text{rd}}\) degree Taylor polynomials of the following functions.

    1. \(f(x)=2x^3-x^2+4x+5\) at \(a=0\)
    2. \(f(x)=\dfrac{1}{1+x}\) at \(a=0\).
    3. \(f(x)=\sqrt{x}\) at \(a=1\)

  5. Derive the Taylor series for \(e^x\) at \(a=0\) (this converges for all \(x\)).

  6. Derive the Taylor series for \(\sin(x)\) at \(a=\dfrac{\pi}{2}\) (this converges for all \(x\)).

  7. From the Taylor series of \(f(x)=\ln(x+1)\) at zero (this converges for \(-1 < x \le 1\)), show that: \[\begin{equation*} \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}=\ln(2). \end{equation*}\]

  8. The Taylor series for \(\tan^{-1}\) at zero is \[\tan^{-1}(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{2n+1}\] which converges for \(|x|\leq 1\). Use this to approximate \(\pi\) (you could do this in MATLAB/Python using a for loop to easily experiment with the number of terms in the sum).

  9. Prove Euler’s formula for complex numbers using the Taylor series for \(e^x\), \(\cos(x)\) and \(\sin(x)\) at \(0\), .

  10. Find the Taylor series for \(\cos(x)\) at zero by differentiating the Taylor series for \(\sin(x)\) at zero term by term (we can differentiate any power series term by term for values of \(x\) where the series converges).

  11. Find the Taylor series of \(\dfrac{1}{1-x}\) at \(a=0\) (this converges for \(|x|<1\)). Hence find the Taylor series of \(\dfrac{1}{(1-x)^2}\) at \(a=0\).