- Find the radius of convergence of the following.
- ∑
_{k=1}^{∞}^{n} - ∑
_{k=1}^{∞}sin3^{n}x^{n} - ∑
_{k=0}^{∞}k!x^{k} - ∑
_{n=0}^{∞}x^{n} - ∑
_{n=0}^{∞}x^{n}

- ∑
- Find ∑
_{k=1}^{∞}k2^{−k}. - Find ∑
_{k=1}^{∞}k^{2}3^{−k}. - Find ∑
_{k=1}^{∞}. - Find ∑
_{k=1}^{∞}. - Show there exists a function f which is continuous on but nowhere differentiable and an infinite series of the form ∑
_{k=1}^{∞}p_{k}where each p_{k}is a polynomial which converges uniformly to fon. Thus it makes absolutely no sense to write something like f^{′}= ∑_{k=1}^{∞}p_{k}^{′}. Hint: Use the Weierstrass approximation theorem. - Find the power series centered at 0 for the function 1∕and give the radius of convergence. Where does the function make sense? Where does the power series equal the function?
- Find a power series for the function f≡for x > 0. Where does fmake sense? Where does the power series you found converge?
- Use the power series technique which was applied in Example 8.4.1 to consider the
initial value problem y
^{′}= y,y= 1 . This yields another way to obtain the power series for e^{x}. - Use the power series technique on the initial value problem y
^{′}+ y = 0,y= 1 . What is the solution to this initial value problem? - Use the power series technique to find solutions in terms of power series to the initial
value problem
Tell where your solution gives a valid description of a solution for the initial value problem. Hint: This is a little different but you proceed the same way as in Example 8.4.1. The main difference is you have to do two differentiations of the power series instead of one.

- Find several terms of a power series solution to the nonlinear initial value
problem
This is the equation which governs the vibration of a pendulum. Explain why there exists a power series which gives the solution to the above initial value problem. Multiply the equation by y

^{′}and identify what you have obtained as the derivative of an interesting quantity which must be constant. - Suppose the function e
^{x}is defined in terms of a power series, e^{x}≡∑_{k=0}^{∞}. Use Theorem 5.4.8 on Page 241 to show directly the usual law of exponents,Be sure to check all the hypotheses.

- Let f
_{n}≡^{1∕2}. Show that for all x,Thus these approximate functions converge uniformly to the function f

=. Now show f_{n}^{′}= 0 for all n and so f_{n}^{′}→ 0. However, the function f≡has no derivative at x = 0. Thus even though f_{n}→ ffor all x, you cannot say that f_{n}^{′}→ f^{′}. - Let the functions, f
_{n}be given in Problem 14 and considerShow that for all x,

and that g

_{k}^{′}= 0 for all k. Therefore, you can’t differentiate the series term by term and get the right answer^{3}. - Use the theorem about the binomial series to give a proof of the binomial
theorem
whenever n is a positive integer.

- Find the power series for sinby plugging in x
^{2}where ever there is an x in the power series for sinx. How do you know this is the power series for sin? - Find the first several terms of the power series for sin
^{2}by multiplying the power series for sin. Next use the trig. identity, sin^{2}=and the power series for costo find the power series. - Find the power series for f=.
- Let a,b be two positive numbers and let p > 1. Choose q such that
Now verify the important inequality

Hint: You might try considering f

=+− ab for fixed b > 0 and examine its graph using the derivative. - Using Problem 20, show that if α > 0,p > 1, it follows that for all x > 0
- Using Problem 21, define for p > 1 and α > 0 the following sequence
Show lim

_{n→∞}x_{n}= x where x = α^{1∕p}. In fact show that after x_{1}the sequence decreases to α^{1∕p}. - Consider the sequence
_{n=1}^{∞}where x is a positive number. Using the binomial theorem show this sequence is increasing. Next show the sequence converges. - Consider the sequence
_{n=1}^{∞}where x is a positive number. Show this sequence decreases when x < 2. Hint: You might consider showing^{(x∕y) +1}is increasing in y provided x ≤ 2. To do this, you might use the following observation repeatedly. If f= 0 and f^{′}> 0, then f≥ 0. There may also be other ways to do this.

Download PDFView PDF