170 CHAPTER 8. POWER SERIES

15. De Moivre’s theorem says [r (cos t + isin t)]n = rn (cosnt + isinnt) for n a positiveinteger. Prove this formula by induction. Does this formula continue to hold for allintegers n, even negative integers? Explain.

16. Using De Moivre’s theorem, show that if z ∈ C then z has n distinct nth roots. Hint:Letting z = x+ iy, z = |z|

(x|z| + i y

|z|

)and argue

(x|z| ,

y|z|

)is a point on the unit circle.

Hence z = |z|(cos(θ)+ isin(θ)) . Then w = |w|(cos(α)+ isin(α)) is an nth root ifand only if (|w|(cos(α)+ isin(α)))n = z. Show this happens exactly when |w| =n√|z| and α = θ+2kπ

n for k = 0,1, · · · ,n.

17. Using De Moivre’s theorem from Problem 15, derive a formula for sin(5x) and onefor cos(5x).

18. Suppose ∑∞n=0 an (x− c)n is a power series with radius of convergence r. Show the

series converge uniformly on any interval [a,b] where [a,b]⊆ (c− r,c+ r) .

19. Find the disc of convergence of the series ∑xn

np for various values of p. Hint: UseDirichlet’s test.

20. Show ex = ∑∞k=0

xk

k! for all x ∈ R where e is the number such that lne = 1. Thus e =∑

∞k=0

1k! . Show e is irrational. Hint: If e = p/q for p,q positive integers, then argue

q!(

pq −∑

qk=0

1k!

)is an integer. However, you can also show q!

(∑

∞k=0

1k! −∑

qk=0

1k!

)<

1

21. Let a≥ 1. Show that for all x > 0, you have the inequality ax > ln(1+ xa) .

8.8 L’Hôpital’s RuleThere is an interesting rule which is often useful for evaluating difficult limits. It is calledL’Hôpital’s 1 rule. The best versions of this rule are based on the Cauchy Mean valuetheorem, Theorem 7.8.2 on Page 144.

Theorem 8.8.1 Let [a,b]⊆ [−∞,∞] and suppose f ,g are functions which satisfy,

limx→b−

f (x) = limx→b−

g(x) = 0, (8.19)

and f ′ and g′ exist on (a,b) with g′ (x) ̸= 0 on (a,b). Suppose also that

limx→b−

f ′ (x)g′ (x)

= L. (8.20)

Then

limx→b−

f (x)g(x)

= L. (8.21)

1L’Hôpital published the first calculus book in 1696. This rule, named after him, appeared in this book. Therule was actually due to Bernoulli who had been L’Hôpital’s teacher. L’Hôpital did not claim the rule as his ownbut Bernoulli accused him of plagarism. Nevertheless, this rule has become known as L’Hôpital’s rule ever since.The version of the rule presented here is superior to what was discovered by Bernoulli and depends on the Cauchymean value theorem which was found over 100 years after the time of L’Hôpital.