- Verify the power series claimed in the chapter for cos,sinand e
^{x}. The method for doing this was shown in the chapter in the case of e^{x}. Go through the details carefully and then do the same details for cos,sin. - In each of the following, assume the relation defines y as a function of x for
values of x and y of interest and find y
^{′}.- xy
^{2}+ sin= x^{3}+ 1 - y
^{3}+ xcos= x^{4} - y cos= tancos+ 2
^{6}= x^{3}y + 3- + cos= 7
- sin= 3 x
- y
^{3}sin+ y^{2}x^{2}= 2^{x2 }y + ln - y
^{2}sinx + log_{3}= y^{2}+ 11 - sin+ sec= e
^{x+y}+ y2^{y}+ 2 - sin+ y
^{3}= 16 - cos+ ln= x
^{2}y + 3

- xy
- In each of the following, assume the relation defines y as a function of x for values
of x and y of interest. Use the chain rule to show y satisfies the given differential
equation.
- x
^{2}y + siny = 7,y^{′}+ 2xy = 0. - x
^{2}y^{3}+ sin= 5 , 2xy^{3}+y^{′}= 0. - y
^{2}sin+ xy = 6,

- x
- Show that if D⊆ U ⊆ D, and if f and g are both one to one, then f ∘ g is also one to one.
- The logarithm test states the following. Suppose a
_{k}≠0 for large k and that p = lim_{k→∞}exists. If p > 1, then ∑_{k=1}^{∞}a_{k}converges absolutely. If p < 1, then the series, ∑_{k=1}^{∞}a_{k}does not converge absolutely. Prove this theorem. - Suppose f= f+ fand f is continuous at 0. Find all solutions to this functional equation which are continuous at x = 0. Now find all solutions which are bounded near 0.
- Suppose f= ffand f is differentiable and not identically zero. Find all solutions to this functional equation. Hint: First show the functional equation requires f > 0.
- Suppose f= f+ ffor x,y > 0. Suppose also f is differentiable. Find all solutions to this functional equation.
- Using the Cauchy condensation test, determine the convergence of ∑
_{k=2}^{∞}. Now determine the convergence of ∑_{k=2}^{∞}. - Find the values of p for which the following series converges and the values of p for
which it diverges.
- For p a positive number, determine the convergence of
for various values of p.

- Determine whether the following series converge absolutely, conditionally, or not at
all and give reasons for your answers.
- ∑
_{n=1}^{∞}^{n} - ∑
_{n=1}^{∞}^{n} - ∑
_{n=1}^{∞}^{n} - ∑
_{n=1}^{∞}^{n}sin - ∑
_{n=1}^{∞}^{n}tan - ∑
_{n=1}^{∞}^{n}cos - ∑
_{n=1}^{∞}^{n}sin

- ∑
- This problem is for people who know about complex numbers. Recall that i
^{2}= −1 and complex numbers are of the form x + iy for x,y real. De Moivre’s theorem saysfor n a positive integer. Prove this formula by induction. Does this formula continue to hold for all integers, n, even negative integers? Explain.

- This is a continuation of the above problem and is only for people who know about
complex numbers. Using De Moivre’s theorem, show that if z ∈ ℂ then z has n
distinct n
^{th}roots. Hint: Letting z = x + iy,and argue

is a point on the unit circle. Hence z =. Thenis an n

^{th}root if and only if^{n}= z. Show this happens exactly when=and α =for k = 0,1,,n. - Using De Moivre’s theorem from Problem 13, derive a formula for sinand one for cos.
- Suppose ∑
_{n=0}^{∞}a_{n}^{n}is a power series with radius of convergence r. Show the series converge uniformly on any intervalwhere⊆. This is in the text but go through the details yourself. - In this problem, x will be a complex number. Thus you will find the disk of
convergence, not just an interval of convergence. In other words, you will find all
complex numbers such that the given series converges. Find the disc of
convergence of the series ∑
for various values of p. Hint: Use Dirichlet’s test.
- The power series for e
^{x}was given above. ThusShow e is irrational. Hint: If e = p∕q for p,q positive integers, then argue

is an integer. However, you can also show

- Let a ≥ 1. Show that for all x > 0, you have the inequality
- Show
Now use this to find a series which converges to arctan

= π∕4. RecallFor which values of x will your series converge? For which values of x does the above description of arctan in terms of an integral make sense? Does this help to show the inferiority of power series?

- Show
Now use the binomial theorem to find a power series for arcsin

.

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